{"id":21,"date":"2023-09-17T11:08:03","date_gmt":"2023-09-17T11:08:03","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e6%97%a0%e7%a9%b7%e5%a4%a7%e7%9a%84%e6%9e%81%e9%99%90\/"},"modified":"2023-09-17T11:08:03","modified_gmt":"2023-09-17T11:08:03","slug":"%e6%97%a0%e7%a9%b7%e5%a4%a7%e7%9a%84%e6%9e%81%e9%99%90","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e6%97%a0%e7%a9%b7%e5%a4%a7%e7%9a%84%e6%9e%81%e9%99%90\/","title":{"rendered":"\u65e0\u7a77\u5927\u7684\u6781\u9650"},"content":{"rendered":"<p>\u5728\u8fd9\u91cc\uff0c\u60a8\u5c06\u4e86\u89e3\u5982\u4f55\u6c42\u89e3\u6240\u6709\u7c7b\u578b\u7684\u65e0\u7a77\u5927\u6781\u9650\uff1a\u591a\u9879\u5f0f\u3001\u6709\u7406\u51fd\u6570\u3001\u6307\u6570\u51fd\u6570\u3001\u6709\u6839\u51fd\u6570\u3001\u65e0\u7a77\u5927\u4e0d\u786e\u5b9a\u6027&#8230;\u6b64\u5916\uff0c\u60a8\u5c06\u80fd\u591f\u901a\u8fc7 25 \u4e2a\u7ec3\u4e60\u8fdb\u884c\u8bad\u7ec3\uff0c\u9010\u6b65\u89e3\u51b3 x \u65f6\u7684\u6781\u9650\u95ee\u9898\u8d8b\u4e8e\u65e0\u9650\u3002 \u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"limite-de-una-funcion-cuando-x-tiende-a-infinito\"><\/span>\u5f53x\u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\u51fd\u6570\u7684\u6781\u9650<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>\u5f53 x \u63a5\u8fd1\u65e0\u7a77\u5927\u65f6\uff0c\u51fd\u6570\u7684\u6781\u9650<\/strong>\uff08\u65e0\u8bba\u662f\u6b63\u6570\u8fd8\u662f\u8d1f\u6570\uff09\u53ef\u4ee5\u662f\u5b9e\u503c\u3001\u6b63\u65e0\u7a77\u5927\u3001\u8d1f\u65e0\u7a77\u5927\u6216\u4e0d\u5b58\u5728\u3002\u8981\u6c42\u89e3\u65e0\u7a77\u5927\u6781\u9650\uff0c\u60a8\u9700\u8981\u5c06 x \u66ff\u6362\u4e3a\u65e0\u7a77\u5927\u3002 <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/limites-a-linfini.webp\" alt=\"\u65e0\u7a77\u5927\u7684\u6781\u9650\" class=\"wp-image-1213\" width=\"601\" height=\"435\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p>\u4ece\u7b2c\u4e00\u5f20\u56fe\u4e2d\u53ef\u4ee5\u770b\u51fa\uff0c\u663e\u793a\u7684\u51fd\u6570\u503e\u5411\u4e8e\u5c06\u5b9e\u9645\u503c<em>k<\/em>\u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\uff0c\u56e0\u4e3a\u968f\u7740<em>x<\/em>\u7684\u589e\u957f\uff0c\u5b83\u4f1a\u8d8a\u6765\u8d8a\u63a5\u8fd1<em>k<\/em> \u3002\u5f53<em>x<\/em>\u63a5\u8fd1\u65e0\u7a77\u5927\u65f6\uff0c\u53f3\u4e0a\u89d2\u7684\u51fd\u6570\u8d8b\u4e8e\u65e0\u7a77\u5927\uff0c\u56e0\u4e3a\u5b83\u968f\u7740<em>x<\/em>\u503c\u7684\u589e\u52a0\u800c\u65e0\u9650\u589e\u957f\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5de6\u4e0b\u89d2\u7684\u56fe\u5f62\u4e0d\u65ad\u51cf\u5c0f\uff0c\u56e0\u6b64\u8d8b\u4e8e\u8d1f\u65e0\u7a77\u5927\u3002\u6700\u540e\uff0c\u6700\u540e\u4e00\u4e2a\u51fd\u6570\u662f\u5468\u671f\u6027\u7684\u5e76\u4e14\u4e0d\u8d8b\u4e8e\u4efb\u4f55\u503c\uff0c\u56e0\u6b64\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u6ca1\u6709\u65e0\u7a77\u5927\u7684\u9650\u5236\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-resolver-limites-al-infinito\"><\/span>\u5982\u4f55\u89e3\u65e0\u7a77\u8fdc\u6781\u9650<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\">\u4e3a\u4e86\u89e3\u51b3\u591a\u9879\u5f0f\u51fd\u6570\u4e2d\u65e0\u7a77\u5927\u7684\u6781\u9650\uff0c\u6211\u4eec\u5fc5\u987b\u4ec5\u5728\u51fd\u6570\u7684\u6700\u9ad8\u6b21\u9879\u4e2d\u5c06 x \u66ff\u6362\u4e3a\u65e0\u7a77\u5927\u3002<\/p>\n<\/div>\n<p>\u4f8b\u5982\uff0c\u770b\u770b\u4ee5\u4e0b\u65e0\u7a77\u5927\u6781\u9650\u7684\u8ba1\u7b97\uff0c\u5176\u4e2d\u6211\u4eec\u4ec5\u5c06\u65e0\u7a77\u5927\u66ff\u6362\u4e3a\u6700\u9ad8\u6b21\u7684\u5355\u9879\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-effad2986ba2e74fbe500e289a69da9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}(3x^2-4x+6) = 3(+\\infty)^2 = \\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"298\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p>\u6b63\u5982\u60a8\u5728\u793a\u4f8b\u4e2d\u6240\u770b\u5230\u7684\uff0c+\u221e \u5e73\u65b9\u7ed9\u51fa +\u221e\uff0c\u56e0\u4e3a\u975e\u5e38\u5927\u7684\u6570 (+\u221e) \u7684 2 \u6b21\u65b9\u59cb\u7ec8\u4f1a\u7ed9\u51fa\u975e\u5e38\u5927\u7684\u6570 (+\u221e)\u3002<\/p>\n<p>\u4e58\u6cd5\u4e5f\u4f1a\u53d1\u751f\u540c\u6837\u7684\u4e8b\u60c5\uff1a\u5982\u679c\u4f60\u4e58\u4ee5\u4e00\u4e2a\u975e\u5e38\u5927\u7684\u6570\uff08+\u65e0\u7a77\u5927\uff09\uff0c\u4f60\u603b\u662f\u4f1a\u5f97\u5230\u4e00\u4e2a\u975e\u5e38\u5927\u7684\u6570\uff08+\u65e0\u7a77\u5927\uff09\u3002\u4f8b\u5982\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d1c763b44697322fd24e76bfa51f5c5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3\\cdot (+\\infty)= +\\infty.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div style=\"padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p><strong>\u8b66\u544a\uff1a<\/strong>\u8981\u8ba1\u7b97\u65e0\u7a77\u5927\u7684\u6781\u9650\uff0c\u8003\u8651\u4ee5\u4e0b\u56e0\u7d20\u5f88\u91cd\u8981\uff1a<\/p>\n<p> <span style=\"color:#FF9B28;\"><strong>\u2192<\/strong><\/span>\u8d1f\u6570\u7684\u5076\u6570\u6307\u6570\u4e3a\u6b63\u6570\u3002\u56e0\u6b64\uff0c\u8d1f\u65e0\u7a77\u5927\u5316\u4e3a\u5076\u6570\u6307\u6570\u5f97\u5230\u6b63\u65e0\u7a77\u5927\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-05dc9f255893d95b9e79b7b5d51dd22e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-\\infty)^2 = +\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#FF9B28;\"><strong>\u2192<\/strong><\/span>\u8d1f\u6570\u7684\u5947\u6570\u6307\u6570\u4e3a\u8d1f\u6570\u3002\u56e0\u6b64\uff0c\u8d1f\u65e0\u7a77\u5927\u5347\u5230\u5947\u6570\u6307\u6570\u5c31\u662f\u8d1f\u65e0\u7a77\u5927\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-49e1e0668bc635216d447fffa91818f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-\\infty)^3 = -\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#FF9B28;\"><strong>\u2192<\/strong><\/span>\u4e58\u4ee5\u8d1f\u6570\u4f1a\u6539\u53d8\u65e0\u7a77\u5927\u7684\u7b26\u53f7\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-74eb011f930bf861413a1f1b76504d87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2(+\\infty) = - \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#FF9B28;\"><strong>\u2192<\/strong><\/span>\u4efb\u4f55\u6570\u5b57\u9664\u4ee5<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1e47b723e734ab9ab0854874654472fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pm \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u7ed9\u51fa 0\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2a3e55d57fa71b3742567248df7ec299_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{5}{\\infty} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"50\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-limites-al-infinito\"><\/span>\u65e0\u7a77\u5927\u6781\u9650\u7684\u4f8b\u5b50<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u56e0\u6b64\uff0c\u60a8\u53ef\u4ee5\u770b\u5230\u5982\u4f55\u5728\u591a\u9879\u5f0f\u4e2d\u89e3\u51b3\u65e0\u7a77\u5927\u7684\u9650\u5236\uff0c\u4ee5\u4e0b\u662f\u89e3\u51b3\u7684\u51e0\u4e2a\u6b64\u7c7b\u9650\u5236\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bbab608d243555490569fab22938c6e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle \\lim_{x \\to +\\infty} (x^3-x^2+4)= (+\\infty) ^3 = \\bm{+\\infty}\\\\[4ex]\\displaystyle\\lim_{x \\to +\\infty} (-5x+2)= -5(+\\infty)= \\bm{-\\infty}\\\\[4ex]\\displaystyle \\lim_{x \\to -\\infty} (x^2-7x+1) = (-\\infty)^2 = \\bm{+\\infty}\\\\[4ex]\\displaystyle \\lim_{x \\to -\\infty} (x^3-x^2+4)= (-\\infty) ^3 = \\bm{-\\infty}\\\\[4ex]\\displaystyle \\lim_{x \\to +\\infty} \\ \\cfrac{1}{x}= \\cfrac{1}{+\\infty} = \\bm{0}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"255\" width=\"280\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"limites-al-infinito-indeterminados\"><\/span>\u672a\u786e\u5b9a\u7684\u65e0\u7a77\u5927\u6781\u9650<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u65e0\u7a77\u5927\u7684\u6781\u9650\u5e76\u4e0d\u603b\u662f\u90a3\u4e48\u5bb9\u6613\u8ba1\u7b97\uff0c\u56e0\u4e3a\u6709\u65f6\u6211\u4eec\u4f1a\u5f97\u5230\u65e0\u7a77\u5927\u4e4b\u95f4\u7684\u65e0\u7a77\u5927\u7684\u4e0d\u786e\u5b9a\u6027\u6216\u65e0\u7a77\u5927\u51cf\u53bb\u65e0\u7a77\u5927\u7684\u4e0d\u786e\u5b9a\u6027\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6d019f26dd82d4b42553a1594f23c061_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\infty}{\\infty}\\qquad \\qquad \\infty-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"145\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u5f53\u6211\u4eec\u5f97\u5230\u8fd9\u79cd\u7c7b\u578b\u7684\u4e0d\u786e\u5b9a\u6027\uff08\u6216\u4e0d\u786e\u5b9a\u5f62\u5f0f\uff09\u65f6\uff0c\u6211\u4eec\u65e0\u6cd5\u76f4\u63a5\u77e5\u9053\u7ed3\u679c\uff0c\u800c\u662f\u5fc5\u987b\u6267\u884c\u521d\u6b65\u7a0b\u5e8f\u6765\u627e\u5230\u6781\u9650\u503c\u3002\u7136\u540e\u6211\u4eec\u5c06\u770b\u5230\u5982\u4f55\u89e3\u51b3\u65e0\u7a77\u8fdc\u7684\u4e0d\u786e\u5b9a\u6781\u9650\u3002 <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-infinito-entre-infinito\"><\/span>\u65e0\u9650\u4e4b\u95f4\u7684\u65e0\u9650\u4e0d\u786e\u5b9a\u6027<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u8981\u627e\u5230\u4e0d\u786e\u5b9a\u6027\u65e0\u7a77\u5927\u9664\u4ee5\u65e0\u7a77\u5927\u7684\u7ed3\u679c\uff0c\u6211\u4eec\u5fc5\u987b\u6bd4\u8f83\u5206\u6570\u7684\u5206\u5b50\u6b21\u6570\u548c\u5206\u6bcd\u6b21\u6570\uff1a<\/p>\n<ol style=\"color:#FF8A05; font-weight: bold;border:\">\n<li style=\"margin-bottom:20px\"><span style=\"color:#101010;font-weight: normal;\">\u5982\u679c\u5206\u5b50\u591a\u9879\u5f0f\u7684\u6b21\u6570\u5c0f\u4e8e\u5206\u6bcd\u591a\u9879\u5f0f\u7684\u6b21\u6570\uff0c\u5219\u65e0\u7a77\u5927\u4e0a\u7684\u65e0\u9650\u4e0d\u786e\u5b9a\u6027<strong><u style=\"text-decoration-color:#FF9B28;\">\u7b49\u4e8e\u96f6\u3002<\/u><\/strong><\/span><\/li>\n<li style=\"margin-bottom:20px\"><span style=\"color:#101010;font-weight: normal;\">\u5982\u679c\u5206\u5b50\u591a\u9879\u5f0f\u7684\u6b21\u6570\u7b49\u4e8e\u5206\u6bcd\u591a\u9879\u5f0f\u7684\u6b21\u6570\uff0c\u5219\u65e0\u7a77\u5927\u4e0a\u7684\u65e0\u9650\u4e0d\u786e\u5b9a\u6027\u662f<strong><u style=\"text-decoration-color:#FF9B28;\">\u4e24\u4e2a\u591a\u9879\u5f0f\u7684\u4e3b\u7cfb\u6570\u7684\u5546\u3002<\/u><\/strong><\/span><\/li>\n<li><span style=\"color:#101010;font-weight: normal;\">\u5982\u679c\u5206\u5b50\u591a\u9879\u5f0f\u7684\u6b21\u6570\u5927\u4e8e\u5206\u6bcd\u591a\u9879\u5f0f\u7684\u6b21\u6570\uff0c\u5219\u65e0\u7a77\u5927\u4e4b\u95f4\u7684\u65e0\u9650\u4e0d\u786e\u5b9a\u6027\u7ed9\u51fa<strong><u style=\"text-decoration-color:#FF9B28;\">\u6216\u591a\u6216\u5c11\u7684\u65e0\u7a77\u5927<\/u><\/strong>\uff08\u7b26\u53f7\u53d6\u51b3\u4e8e\u4e24\u4e2a\u591a\u9879\u5f0f\u7684\u4e3b\u8981\u9879\uff09\u3002<\/span><\/li>\n<\/ol>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c969e4b99985b44006e57d554ff0247_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to \\pm \\infty}}\\frac{a_nx^r+a_{n-1}x^{r-1}+a_{n-2}x^{r-2}+\\dots}{b_nx^s+b_{n-1}x^{s-1}+b_{n-2}x^{s-2}+\\dots}=\\left\\{ \\begin{array}{lcl} 0 &amp; \\text{si} &amp; r<s \\\\[3ex]=&quot;&quot; \\cfrac{a_n}{b_n}=&quot;&quot; &amp;=&quot;&quot; \\text{si}=&quot;&quot; r=&quot;s&quot; \\\\[5ex]=&quot;&quot; \\pm=&quot;&quot; \\infty=&quot;&quot;>s \\end{array}\\right.&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;139&#8243; width=&#8221;767&#8243; style=&#8221;vertical-align: 0px;&#8221;><\/p>\n<\/p>\n<p>\u4f8b\u5982\uff0c\u5728\u4e0b\u9762\u7684\u6781\u9650\u4e2d\uff0c\u5206\u5b50\u591a\u9879\u5f0f\u662f\u4e8c\u6b21\u591a\u9879\u5f0f\uff0c\u800c\u5206\u6bcd\u591a\u9879\u5f0f\u662f\u4e09\u6b21\u591a\u9879\u5f0f\uff0c\u56e0\u6b64\u6781\u9650\u7684\u89e3\u4e3a 0\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b4e5e0058ab08d743a6dc18587912a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{6x^2-5}{x^3+1} = \\cfrac{6(+\\infty)^2}{(+\\infty)^3} = \\cfrac{+\\infty}{+\\infty}= \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"293\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u770b\u53e6\u4e00\u4e2a\u4f8b\u5b50\uff0c\u5176\u4e2d\u6709\u7406\u51fd\u6570\u7684\u4e24\u4e2a\u591a\u9879\u5f0f\u90fd\u662f\u4e8c\u9636\u7684\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u9664\u4ee5\u66f4\u9ad8\u9636\u9879\u7684\u7cfb\u6570\u624d\u80fd\u8ba1\u7b97\u65e0\u7a77\u5927\u7684\u6781\u9650\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c21a5f7720fd6be40b043d30f904941_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{4x^2+1}{2x^2-5} = \\cfrac{4(+\\infty)^2}{2(+\\infty)^2}= \\cfrac{+\\infty}{+\\infty} =\\cfrac{4}{2} = \\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"327\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u6700\u540e\uff0c\u5728\u4e0b\u4e00\u4e2a\u6781\u9650\u5904\uff0c\u5206\u5b50\u7684\u51fd\u6570\u7684\u6b21\u6570\u5927\u4e8e\u5206\u6bcd\u7684\u51fd\u6570\u7684\u6b21\u6570\uff0c\u56e0\u6b64\u65e0\u7a77\u5927\u5bf9\u65e0\u7a77\u5927\u7684\u4e0d\u786e\u5b9a\u6027\u7ed9\u51fa\u65e0\u7a77\u5927\u3002\u53e6\u5916\uff0c\u4ece\u5206\u5b50\u5f97\u5230\u6b63\u65e0\u7a77\u5927\uff0c\u4ece\u5206\u6bcd\u5f97\u5230\u8d1f\u65e0\u7a77\u5927\uff0c\u6240\u4ee5\u6781\u9650\u7684\u7ed3\u679c\u662f\u8d1f\u6570\uff08\u8d1f\u6570\u4e4b\u95f4\u7684\u6b63\u6570\u662f\u8d1f\u6570\uff09\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-de6d4de74f4fe69e45ce1a55fcb8c7d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} \\cfrac{3x^2+2x-5}{7x+1} = \\cfrac{3(-\\infty)^2}{7(-\\infty)}=\\cfrac{3(+\\infty)}{-\\infty}}= \\cfrac{+\\infty}{-\\infty}= \\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"436\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h4 class=\"wp-block-heading\">\u65e0\u7a77\u5927\u4e0e\u6839\u4e4b\u95f4\u7684\u65e0\u9650\u4e0d\u786e\u5b9a\u6027<\/h4>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c<strong>\u65e0\u7406\u51fd\u6570\uff08\u6709\u6839\u51fd\u6570\uff09\u7684\u6b21\u6570<\/strong>\u662f\u4e3b\u9879\u7684\u6b21\u6570\u4e0e\u6839\u5f0f\u7d22\u5f15\u4e4b\u95f4\u7684\u5546\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ffc00917d2cc316211a57feafdddd0d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt[\\color{red}\\bm{m}\\color{black}]{a_nx^{\\color{blue}\\bm{n}\\color{black}}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\\dots} \\ \\longrightarrow \\ \\text{grado}=\\cfrac{\\color{blue}\\bm{n}\\color{black}}{\\color{red}\\bm{m}\\color{black}}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"580\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u5982\u679c<strong>\u6709\u6839\u51fd\u6570\u7684\u6781\u9650\u7ed9\u51fa infinity \u4e4b\u95f4\u7684\u65e0\u9650\u4e0d\u786e\u5b9a\u6027<\/strong>\uff0c\u6211\u4eec\u5fc5\u987b\u5e94\u7528\u4e0a\u9762\u89e3\u91ca\u7684\u5173\u4e8e\u5206\u5b50\u548c\u5206\u6bcd\u7684\u6b21\u6570\u7684\u76f8\u540c\u89c4\u5219\uff0c\u4f46\u8981\u8003\u8651\u5230\u6709\u6839\u591a\u9879\u5f0f\u7684\u6b21\u6570\u7684\u8ba1\u7b97\u65b9\u5f0f\u4e0d\u540c\u3002<\/p>\n<p>\u770b\u4e0b\u9762\u5e26\u6709\u6839\u5f0f\u7684\u51fd\u6570\u7684\u65e0\u9650\u6781\u9650\u793a\u4f8b\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b93ef0d623e6904538b361f5d6f1ef9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{4x^2+11}{\\sqrt{x^8-3x^2-5}}=\\frac{4(+\\infty)^2}{\\sqrt{(+\\infty)^8}}=\\frac{+\\infty}{+\\infty}=\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"354\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p>\u5206\u5b50\u7684\u6b21\u6570\u4e3a 2\uff0c\u5206\u6bcd\u7684\u6b21\u6570\u4e3a 4 (8\/2=4)\uff0c\u56e0\u6b64\u6781\u9650\u4e3a 0\uff0c\u56e0\u4e3a\u5206\u5b50\u7684\u6b21\u6570\u5c0f\u4e8e\u5206\u6bcd\u7684\u6b21\u6570\u3002<\/p>\n<h4 class=\"wp-block-heading\">\u65e0\u7a77\u5927\u4e0e\u6307\u6570\u51fd\u6570\u4e4b\u95f4\u7684\u65e0\u9650\u4e0d\u786e\u5b9a\u6027<\/h4>\n<p>\u6307\u6570\u51fd\u6570\u7684\u589e\u957f\u8fdc\u5927\u4e8e\u591a\u9879\u5f0f\u51fd\u6570\u7684\u589e\u957f\uff0c<strong>\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u8003\u8651\u6307\u6570\u51fd\u6570\u7684\u6b21\u6570\u5927\u4e8e\u591a\u9879\u5f0f\u51fd\u6570\u7684\u6b21\u6570\u3002<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-49d708f83c6876b3cdb6d884ab7b6a23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{exponencial}>\\text{polinomio}&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;16&#8243; width=&#8221;192&#8243; style=&#8221;vertical-align: -4px;&#8221;><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u5982\u679c\u4e0d\u786e\u5b9a\u6027\u65e0\u7a77\u5927\u9664\u4ee5\u65e0\u7a77\u5927\u662f\u7531\u6307\u6570\u51fd\u6570\u7684\u6781\u9650\u5f97\u51fa\u7684\uff0c\u6211\u4eec\u5fc5\u987b\u7b80\u5355\u5730\u5e94\u7528\u89e3\u91ca\u5206\u5b50\u548c\u5206\u6bcd\u7684\u6b21\u6570\u7684\u76f8\u540c\u89c4\u5219\uff0c\u4f46\u8981\u8003\u8651\u5230\u6307\u6570\u51fd\u6570\u7684\u9636\u6570\u6bd4\u591a\u9879\u5f0f\u9ad8\u3002<\/p>\n<p>\u6b64\u5916\uff0c\u5982\u679c\u9664\u6cd5\u7684\u5206\u5b50\u548c\u5206\u6bcd\u4e2d\u6709\u6307\u6570\u51fd\u6570\uff0c\u5219\u5177\u6709\u6700\u5927\u57fa\u6570\u7684\u6307\u6570\u51fd\u6570\u5c06\u662f\u6700\u9ad8\u9636\u7684\u6307\u6570\u51fd\u6570\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-50f9e93066ce9e76b76ef6c7a72a9fad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{7x^5+6x^3-4x}{4^x}=\\frac{7(+\\infty)^5}{4^{+\\infty}}=\\frac{+\\infty}{+\\infty}=\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"350\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p>\u5728\u6b64\u793a\u4f8b\u4e2d\uff0c\u5206\u6bcd\u7531\u6307\u6570\u51fd\u6570\u6784\u6210\uff0c\u56e0\u6b64\u5b83\u7684\u9636\u6570\u6bd4\u5206\u5b50\u9ad8\u3002\u56e0\u6b64\uff0c\u65e0\u7a77\u5927\u4e4b\u95f4\u7684\u4e0d\u5b9a\u5f62\u5f0f\u65e0\u7a77\u5927\u7ed9\u51fa0\u3002 <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-infinito-menos-infinito\"><\/span>\u65e0\u7a77\u8d1f\u65e0\u7a77\u4e0d\u786e\u5b9a\u6027<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u6c42\u89e3\u65e0\u7a77\u51cf\u65e0\u7a77\u4e0d\u786e\u5b9a\u6027\u53d6\u51b3\u4e8e\u51fd\u6570\u662f\u5426\u6709\u5206\u6570\u6216\u6839\u3002\u90a3\u4e48\u8ba9\u6211\u4eec\u770b\u770b\u5982\u4f55\u89e3\u51b3\u8fd9\u4e24\u79cd\u4e0d\u540c\u60c5\u51b5\u7684\u8fd9\u79cd\u4e0d\u786e\u5b9a\u6027\u3002<\/p>\n<h4 class=\"wp-block-heading\">\u4e0d\u786e\u5b9a\u6027 \u65e0\u9650\u51cf\u65e0\u7a77 \u5e26\u5206\u6570<\/h4>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"><strong>\u5f53\u4ee3\u6570\u5206\u6570\u7684\u52a0\u6cd5\u6216\u51cf\u6cd5\u51fa\u73b0\u65e0\u7a77\u8d1f\u65e0\u7a77\u4e0d\u786e\u5b9a\u6027<\/strong>\u65f6\uff0c\u5fc5\u987b\u5148\u8fdb\u884c\u5206\u6570\u7684\u52a0\u6cd5\u6216\u51cf\u6cd5\uff0c\u7136\u540e\u8ba1\u7b97\u6781\u9650\u3002<\/p>\n<\/div>\n<p>\u8ba9\u6211\u4eec\u901a\u8fc7\u9010\u6b65\u6c42\u89e3\u793a\u4f8b\u6765\u4e86\u89e3\u5982\u4f55\u5728\u5e26\u5206\u6570\u7684\u51fd\u6570\u4e2d\u8ba1\u7b97\u4e0d\u786e\u5b9a\u6027\u65e0\u7a77\u5927\u51cf\u53bb\u65e0\u7a77\u5927\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c58eb86af2eb0393a802fc7a29f8a453_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\left( \\frac{x^2}{x-1} - \\frac{x}{3}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"152\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u9996\u5148\u5c1d\u8bd5\u8ba1\u7b97\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b3a2cbbfec28f9de05668b90e9ee65f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\left(  \\frac{x^2}{x-1} - \\frac{x}{3}\\right) = \\frac{(+\\infty)^2}{(+\\infty)-1} - \\frac{+\\infty}{3} = \\bm{+\\infty - \\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"410\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u4f46\u6211\u4eec\u5f97\u5230\u4e86\u4e0d\u786e\u5b9a\u6027 \u221e-\u221e\u3002<\/p>\n<p>\u6211\u4eec\u5fc5\u987b\u9996\u5148\u51cf\u53bb\u5206\u6570\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u5c06\u5206\u6570\u7b80\u5316\u4e3a\u4e00\u4e2a\u516c\u5206\u6bcd\uff0c\u5373\u5c06\u4e00\u4e2a\u5206\u6570\u7684\u5206\u5b50\u548c\u5206\u6bcd\u4e58\u4ee5\u53e6\u4e00\u4e2a\u5206\u6570\u7684\u5206\u6bcd\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-68e489c5833478cb20929ea07ae2971d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x \\to +\\infty} \\left( \\frac{x^2}{x-1}-\\frac{x}{3}\\right)=\\\\[5ex]\\displaystyle =\\lim_{x \\to +\\infty}\\left(\\frac{x^2 \\cdot 3}{(x-1)\\cdot 3}- \\frac{x\\cdot (x-1)}{3\\cdot (x-1)} \\right)=\\\\[5ex]\\displaystyle = \\lim_{x \\to +\\infty} \\left( \\frac{3x^2 }{3(x-1)}- \\frac{x^2-x}{3(x-1)}\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"188\" width=\"295\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u73b0\u5728\u4e24\u4e2a\u5206\u6570\u5177\u6709\u76f8\u540c\u7684\u5206\u6bcd\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b83\u4eec\u5408\u5e76\u4e3a\u4e00\u4e2a\u5206\u6570\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1e5345a6d68ae0cdda543b81f89daa62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\frac{3x^2 -(x^2-x)}{3(x-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"163\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u5bf9\u5206\u5b50\u548c\u5206\u6bcd\u8fdb\u884c\u8fd0\u7b97\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-31cbae0091a641d74250fae5758b3116_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}  \\frac{3x^2 -x^2+x}{3x-3} =  \\lim_{x \\to +\\infty}  \\frac{2x^2+x}{3x-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"284\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p>\u6700\u540e\u6211\u4eec\u518d\u6b21\u8ba1\u7b97\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6ef29c026035a5353b2bada5bc0d9ff9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\frac{2x^2+x}{3x-3}=\\frac{+\\infty}{+\\infty}=\\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"225\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p>\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u65e0\u7a77\u5927\u4e4b\u95f4\u7684\u65e0\u9650\u4e0d\u786e\u5b9a\u6027\u7ed9\u51fa+\u221e\uff0c\u56e0\u4e3a\u5206\u5b50\u7684\u6b21\u6570\u5927\u4e8e\u5206\u6bcd\u7684\u6b21\u6570\u3002<\/p>\n<h4 class=\"wp-block-heading\">\u4e0d\u5b9a\u5f0f\u65e0\u7a77\u51cf\u65e0\u7a77\u6709\u6839<\/h4>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\">\u5f53<strong>\u6839\u5f0f\u52a0\u6cd5\u6216\u6839\u5f0f\u51cf\u6cd5\u51fa\u73b0\u65e0\u7a77\u8d1f\u65e0\u7a77\u4e0d\u786e\u5b9a\u6027<\/strong>\u65f6\uff0c\u5fc5\u987b\u5148\u5c06\u51fd\u6570\u4e58\u4ee5\u3001\u9664\u4ee5\u5171\u8f6d\u6839\u5f0f\u8868\u8fbe\u5f0f\uff0c\u7136\u540e\u6c42\u6781\u9650\u3002<\/p>\n<\/div>\n<p>\u8ba9\u6211\u4eec\u901a\u8fc7\u4ee5\u4e0b\u5206\u6b65\u793a\u4f8b\u6765\u4e86\u89e3\u5982\u4f55\u6c42\u89e3\u65e0\u7406\u51fd\u6570\u4e2d\u7684\u4e0d\u786e\u5b9a\u6027\u65e0\u7a77\u5927\u51cf\u65e0\u7a77\u5927\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e093b62c357684fe8a8818df58d7b99a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(x-\\sqrt{x^2-5}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"165\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u9996\u5148\u5c1d\u8bd5\u6c42\u89e3\u5e26\u6839\u51fd\u6570\u7684\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4459c2b6c968344878499cfbb30adda4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(x-\\sqrt{x^2-5}\\right)=+\\infty-\\sqrt{(+\\infty)^2}=\\bm{+\\infty-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"409\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p>\u7136\u800c\uff0c\u6211\u4eec\u5f97\u5230\u4e86\u4e0d\u5b9a\u5f62\u5f0f \u221e-\u221e\u3002\u56e0\u6b64\uff0c\u8981\u77e5\u9053\u65e0\u7a77\u5927\u51cf\u53bb\u65e0\u7a77\u5927\u6709\u591a\u5c11\u4e0d\u786e\u5b9a\u6027\uff0c\u60a8\u5fc5\u987b\u5e94\u7528\u6240\u89e3\u91ca\u7684\u8fc7\u7a0b\u3002<\/p>\n<p>\u7531\u4e8e\u51fd\u6570\u6709\u6839\u5f0f\uff0c\u6211\u4eec\u5c06\u6574\u4e2a\u51fd\u6570\u4e58\u4ee5\u548c\u9664\u4ee5\u5171\u8f6d\u65e0\u7406\u8868\u8fbe\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f10d91882a0f8dcca86fbb8dda7da7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\left(x-\\sqrt{x^2-5}\\right)= \\lim_{x \\to +\\infty}\\frac{\\left(x-\\sqrt{x^2-5}\\right)\\cdot\\left(x+\\sqrt{x^2-5}\\right)}{x+\\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"488\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u5206\u5b50\u7684\u4ee3\u6570\u8868\u8fbe\u5f0f\u5bf9\u5e94\u4e8e\u548c\u4e0e\u5dee\u7684\u4e58\u79ef\u7684\u663e\u7740\u6052\u7b49\u5f0f\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u7b80\u5316\u8be5\u8868\u8fbe\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b00f177bdb579dabf9dc589e387344cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{\\left(x-\\sqrt{x^2-5}\\right) \\cdot \\left(x + \\sqrt{x^2-5}\\right)}{ x + \\sqrt{x^2-5}}= \\lim_{x \\to +\\infty} \\cfrac{x^2- \\left( \\sqrt{x^2-5}\\right)^2}{ x + \\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"505\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u73b0\u5728\u6211\u4eec\u7b80\u5316\u6781\u9650\u7684\u6839\uff0c\u56e0\u4e3a\u5b83\u662f\u5e73\u65b9\u7684\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d5c798f099ef1c56a50526e7fba8c99c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\frac{x^2-(x^2-5)}{x+\\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u5bf9\u5206\u6570\u7684\u5206\u5b50\u8fdb\u884c\u8fd0\u7b97\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-392ae211b16ad803eb70cc4993a0c7e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\frac{x^2- x^2+5}{x+\\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"146\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-be954eaf609b9f98c6dc984758599b5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\frac{5}{x+\\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"146\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u6700\u540e\uff0c\u6211\u4eec\u91cd\u65b0\u8fdb\u884c\u6781\u9650\u8ba1\u7b97\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c29edfa5eba2fe54e369c3d963d11a45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\frac{5}{x+\\sqrt{x^2-5}}=\\frac{5}{+\\infty+\\sqrt{(+\\infty)^2}}=\\frac{5}{+\\infty}=\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"391\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\u6781\u9650\u7684\u7ed3\u679c\u662f 0\uff0c\u56e0\u4e3a\u4efb\u4f55\u6570\u5b57\u9664\u4ee5\u65e0\u7a77\u5927\u90fd\u7b49\u4e8e 0\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-limites-al-infinito\"><\/span>\u89e3\u51b3\u4e86\u65e0\u7a77\u5927\u6781\u9650\u7684\u7ec3\u4e60<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e601<\/h3>\n<p>\u6c42\u56fe\u51fd\u6570\u7684\u4e0b\u5217\u6781\u9650\uff1a <\/p>\n<div class=\"wp-block-columns are-vertically-aligned-center is-layout-flex wp-container-117\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\" style=\"flex-basis:33.33%\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f78faaaf0015cb381ddcf34bf391f8e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -\\infty}f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"83\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e3ecd9def1bc56849bd20db3e3b0aa1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"83\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e6ff759e69ca5ebf8006e8561f3974d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -1^-}f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"86\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ec06a733cdd869885c350a89160c3e4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -1^+}f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"86\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67d6b9ca176235d3d9293f6631b4ecfa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1^-}f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"75\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-059c38ab3ff7d7f2bf81bda03e9a50fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1^+}f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"75\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\" style=\"flex-basis:66.66%\">\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/fonction-representation-limites-infini.webp\" alt=\"\u4ece\u51fd\u6570\u7684\u8868\u793a\u5230\u65e0\u7a77\u5927\u7684\u6781\u9650\" width=\"401\" height=\"404\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5f53 x \u8d8b\u4e8e\u8d1f\u65e0\u7a77\u5927\u4e14\u6b63\u65e0\u7a77\u5927\u65f6\uff0c\u51fd\u6570\u7684\u6781\u9650\u7ed9\u51fa 1\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce8d3cf96ad1cfddf4436035dc448493_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -\\infty}f(x)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"115\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-85a697eaefeaeed1dc19fd122cf35db9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}f(x)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"115\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u51fd\u6570\u5728 x=-1 \u5904\u7684\u5de6\u53f3\u4fa7\u6781\u9650\u5206\u522b\u4e3a\u6b63\u65e0\u7a77\u5927\u548c\u8d1f\u65e0\u7a77\u5927\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9d42ef8260114e983c7b7ad3fd442b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -1^-}f(x)=+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"141\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-55e81465718a80fe01a65266966d16b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -1^+}f(x)=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"141\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u5f53 x \u8d8b\u4e8e 1 \u65f6\uff0c\u51fd\u6570\u7684\u6a2a\u5411\u6781\u9650\u503c\u662f\u8d1f\u65e0\u7a77\u5927\u548c\u6b63\u65e0\u7a77\u5927\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a24c9f6c0d36f0369628f42d85b00396_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1^-}f(x)=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"131\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-03137c4e19909b490b88ae4b8cb7f27e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1^+}f(x)=+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"130\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e602<\/h3>\n<p>\u5f53 x \u63a5\u8fd1\u4e0b\u5217\u51fd\u6570\u7684\u65e0\u7a77\u5927\u65f6\u6c42\u89e3\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-351efa993cac2aee17802d2bbe17b081_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} (x^2+4x+1)\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"148\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u4e3a\u4e86\u6c42\u89e3\u65e0\u7a77\u5927\u6781\u9650\uff0c\u6211\u4eec\u9700\u8981\u5c06\u591a\u9879\u5f0f\u6700\u9ad8\u6b21\u9879\u4e2d\u7684 x \u66ff\u6362\u4e3a\u65e0\u7a77\u5927\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1c68654644566af566d93d558f974bae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} (x^2+4x+1) = (+\\infty)^2= \\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"280\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e603<\/h3>\n<p>\u8ba1\u7b97\u4ee5\u4e0b\u591a\u9879\u5f0f\u51fd\u6570\u7684\u65e0\u7a77\u5927\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e3ee0c88b6e0c35b635bf70b34fdf007_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} (-3x^2+8x+5)\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"171\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u4e3a\u4e86\u6c42\u89e3\u65e0\u7a77\u5927\u6781\u9650\uff0c\u6211\u4eec\u5c06\u591a\u9879\u5f0f\u6700\u9ad8\u6b21\u9879\u4e2d\u7684 x \u66ff\u6362\u4e3a\u65e0\u7a77\u5927\u5e76\u6267\u884c\u8ba1\u7b97\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bc85257769ef819973ee5ff70f916502_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} (-3x^2+8x+5) = -3(+\\infty)^2= -3\\cdot (+\\infty) = \\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"430\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e604<\/h3>\n<p>\u6c42\u89e3\u4ee5\u4e0b\u591a\u9879\u5f0f\u51fd\u6570\u7684\u81f3\u5c11\u65e0\u9650\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b6deaa6136e2ba9fcf106337c89ea76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} (6x^2-3x-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"157\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u4e3a\u4e86\u8ba1\u7b97\u65e0\u7a77\u5927\u7684\u6781\u9650\uff0c\u6211\u4eec\u7528\u591a\u9879\u5f0f\u6700\u9ad8\u6b21\u9879\u4e2d\u7684\u8d1f\u65e0\u7a77\u5927\u66ff\u6362 x \u5e76\u8ba1\u7b97\u51fd\u6570\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9bf673d8892301f5fd258cdff341d3b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} (6x^2-3x-4) = 6(-\\infty)^2= 6\\cdot (+\\infty) = \\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"388\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u7531\u4e8e\u8d1f\u65e0\u7a77\u5927\u662f\u5e73\u65b9\u7684\uff0c\u56e0\u6b64\u65e0\u7a77\u5927\u7684\u7b26\u53f7\u53d8\u4e3a\u6b63\u6570\u3002<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e605<\/h3>\n<p>\u6c42\u4ee5\u4e0b\u6709\u7406\u51fd\u6570\u7684\u65e0\u7a77\u5927\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-975b04e8343741d4f58e17b8a8d301d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\frac{7}{2x-5}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"101\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u4e3a\u4e86\u786e\u5b9a\u65e0\u7a77\u5927\u7684\u6781\u9650\uff0c\u6211\u4eec\u5c06\u5206\u6570\u5206\u5b50\u548c\u5206\u6bcd\u7684\u6700\u9ad8\u6b21\u6570\u9879\u5904\u7684 x \u66ff\u6362\u4e3a\u6b63\u65e0\u7a77\u5927\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f2b6eb1159801a5793480a1054fa6d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\frac{7}{2x-5} = \\cfrac{7}{2\\cdot(+\\infty)}=\\frac{7}{+\\infty}=\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"287\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u8bf7\u8bb0\u4f4f\uff0c\u4efb\u4f55\u6570\u5b57\u9664\u4ee5\u6b63\u8d1f\u65e0\u7a77\u5927\u90fd\u7b49\u4e8e 0\u3002<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e606<\/h3>\n<p>\u6c42\u89e3\u4ee5\u4e0b\u65e0\u7a77\u8fdc\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-445dcbd97c1aa876a69d4dd05d53e74a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} (-x^3+x^2+5x)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"171\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u8981\u8ba1\u7b97\u51fd\u6570 x \u8d8b\u4e8e \u00b1\u221e \u65f6\u7684\u6781\u9650\uff0c\u53ea\u9700\u67e5\u770b\u51fd\u6570\u6700\u9ad8\u6b21\u6570\u7684\u5355\u9879\u5f0f\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-813779b52206df3a7b3f79e61c8f80b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} (-x^3+x^2+5x) = -(-\\infty)^3= -(-\\infty)= \\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"399\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e607<\/h3>\n<p>\u5f53 x \u63a5\u8fd1\u8d1f\u65e0\u7a77\u5927\u65f6\uff0c\u8ba1\u7b97\u4ee5\u4e0b\u51fd\u6570\u7684\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bf684c38b82f58a5ec523937341266c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} (-4x^2+4)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"130\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u7528\u65e0\u7a77\u5927\u4ee3\u66ff\u4e8c\u6b21\u9879\u5c31\u8db3\u591f\u4e86\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5b84279ebd1a992bf5dc62391a1ae94a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} (-4x^2+4) = -4(-\\infty)^2= -4\\cdot (+\\infty) = \\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"389\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e608<\/h3>\n<p>\u5f53 x \u63a5\u8fd1\u65e0\u7a77\u5927\u65f6\uff0c\u6c42\u4ee5\u4e0b\u6307\u6570\u51fd\u6570\u7684\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3c4f75bf93e725766c276a50c833b31f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} 2^x\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"66\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u867d\u7136\u662f\u6307\u6570\u51fd\u6570\uff0c\u4f46\u6c42\u89e3\u6781\u9650\u7684\u8fc7\u7a0b\u662f\u4e00\u6837\u7684\uff1a\u5c06x\u66ff\u6362\u4e3a\u65e0\u7a77\u5927\u3002 <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a0e42089daf32c26d5360dbdd9fe456c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} 2^x = 2^{+\\infty}=\\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"179\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e609<\/h3>\n<p>\u6c42\u89e3\u4ee5\u4e0b\u6307\u6570\u51fd\u6570\u7684\u65e0\u9650\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0fb53888a185f8feeed50217b2a4536b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} 5^{-x}\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"77\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u8981\u89e3\u51b3\u6b64\u9650\u5236\uff0c\u60a8\u5fc5\u987b\u4f7f\u7528\u5206\u6570\u7684\u5c5e\u6027\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-84a1cfc14727841b3ae54820bfdbb2c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} 5^{-x} = 5^{-(+\\infty)}=5^{-\\infty}= \\cfrac{1}{5^{+\\infty}}= \\cfrac{1}{\\infty} =\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"350\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6010<\/h3>\n<p>\u6c42\u89e3\u4ee5\u4e0b\u65e0\u7a77\u8fdc\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f2459122cc1d9e723b3f78d858c48fe1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{-4x^2+3}{3x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"130\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u8be5\u6781\u9650\u7ed9\u51fa\u4e86\u6b63\u65e0\u7a77\u5927\u4e0e\u8d1f\u65e0\u7a77\u5927\u4e4b\u95f4\u7684\u4e0d\u786e\u5b9a\u6027\u3002\u5206\u5b50\u7684\u6b21\u6570\u5927\u4e8e\u5206\u6bcd\u7684\u6b21\u6570\uff0c\u56e0\u6b64\u4e0d\u5b9a\u6781\u9650\u7b49\u4e8e\u52a0\u4e0a\u65e0\u7a77\u5927\u3002\u7136\u800c\uff0c\u7531\u4e8e\u9664\u6cd5\u662f\u8d1f\u65e0\u7a77\u9664\u4ee5\u6b63\u65e0\u7a77\uff0c\u56e0\u6b64\u7ed3\u679c\u662f\u8d1f\u65e0\u7a77\u3002 <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a446d2cb568ab87f57eb43614c7727e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{-4x^2+3}{3x+1} = \\cfrac{-4(+\\infty)^2}{3(+\\infty)} =\\cfrac{-4(+\\infty)}{+\\infty}= \\cfrac{-\\infty}{+\\infty}= \\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"460\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6011<\/h3>\n<p>\u4fee\u590d\u4ee5\u4e0b\u4e0d\u786e\u5b9a\u9650\u5236\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-08b74c12124842886ef576ef8c4eeb1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{5x+8}{-5x+2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"115\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5728\u8fd9\u4e2a\u95ee\u9898\u4e2d\uff0c\u65e0\u7a77\u5927\u7684\u4e0d\u5b9a\u5f62\u5f0f\u662f\u7531\u4e24\u4e2a\u540c\u6b21\u591a\u9879\u5f0f\u7684\u5546\u5f97\u5230\u7684\uff0c\u56e0\u6b64\u4e0d\u5b9a\u6781\u9650\u7684\u7ed3\u679c\u662f\u5b83\u4eec\u7684\u4e3b\u8981\u7cfb\u6570\u7684\u9664\u6cd5\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc2fb0ed175e50d56e670681c136cd17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{5x+8}{-5x+2} = \\cfrac{5(+\\infty)}{-5(+\\infty)} = \\cfrac{+\\infty}{-\\infty}=\\cfrac{5}{-5}= \\bm{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"367\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6012<\/h3>\n<p>\u8ba1\u7b97\u4ee5\u4e0b\u81f3\u5c11\u5230\u65e0\u7a77\u5927\u7684\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c0431a362c02fce505f4567e28f21fa3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} \\cfrac{x^2+3x+5}{x^4-x-6}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"141\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5206\u5b50\u7684\u4ee3\u6570\u8868\u8fbe\u5f0f\u6b21\u6570\u5c0f\u4e8e\u5206\u6bcd\u7684\u4ee3\u6570\u8868\u8fbe\u5f0f\u6b21\u6570\uff0c\u56e0\u6b64\u4e0d\u786e\u5b9a\u6027 +\u221e\/+\u221e \u7ed9\u51fa 0\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-405fbcd016c064f414b043abe04fa768_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} \\cfrac{x^2+3x+5}{x^4-x-6} = \\cfrac{(-\\infty)^2}{(-\\infty)^4} = \\cfrac{+\\infty}{+\\infty}= \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"316\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6013<\/h3>\n<p>\u6c42\u89e3\u4ee5\u4e0b\u6709\u6839\u51fd\u6570\u7684\u4e0d\u5b9a\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-159a0cb8cc6c1e4551195c4bb03eacd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\sqrt[3]{x^7-4x^3}}{x^2+5x}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"134\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5206\u5b50\u7684\u8868\u8fbe\u5f0f\u4f4d\u4e8e\u6839\u5f0f\u4e0b\uff0c\u56e0\u6b64\u5176\u6b21\u6570\u4e3a 7\/3\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6bcd\u591a\u9879\u5f0f\u662f\u4e8c\u6b21\u7684\u3002\u7531\u4e8e 7\/3&gt;2\uff0c\u6781\u9650\u7ed9\u51fa\u4e86\u66f4\u591a\u7684\u65e0\u7a77\u5927\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7062fdca2096873f9b687699846c27f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\frac{\\sqrt[3]{x^7-4x^3}}{x^2+5x}=\\frac{\\sqrt[3]{(+\\infty)^7}}{(+\\infty)^2}=\\frac{+\\infty}{+\\infty}=\\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"348\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6014<\/h3>\n<p>\u786e\u5b9a\u4ee5\u4e0b\u5e26\u5206\u6570\u51fd\u6570\u7684\u65e0\u9650\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6ffef148096d3aa64a2eb5d63e00d2f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty} \\cfrac{-2x^2}{5-4x}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"101\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5728\u672c\u7ec3\u4e60\u4e2d\uff0c\u6211\u4eec\u5f97\u5230\u4e0d\u786e\u5b9a\u6027\u8d1f\u65e0\u7a77\u5927\u9664\u4ee5\u8d1f\u65e0\u7a77\u5927\uff0c\u5176\u4e2d\u5206\u5b50\u7684\u6b21\u6570\u5927\u4e8e\u5206\u6bcd\u7684\u6b21\u6570\uff0c\u56e0\u6b64\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0b2dfa8a24dd69065fc8ddcf223321d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{-2x^2}{5-4x} = \\cfrac{-2(+\\infty)^2}{-4(+\\infty)} = \\cfrac{-2(+\\infty)}{-\\infty}= \\cfrac{-\\infty}{-\\infty} =\\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"431\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6015<\/h3>\n<p>\u6c42\u4e0b\u5217\u51fd\u6570\u7684\u81f3\u5c11\u65e0\u9650\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-68566303139abd794f304c979271a058_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} \\cfrac{9x}{4-x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"100\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5206\u6bcd\u591a\u9879\u5f0f\u662f\u4e8c\u6b21\u7684\uff0c\u800c\u5206\u5b50\u591a\u9879\u5f0f\u662f\u7ebf\u6027\u7684\u3002\u56e0\u6b64\uff0c\u65e0\u9650\u7684\u4e0d\u786e\u5b9a\u6027\u9664\u4ee5\u65e0\u7a77\u5927\u5f97\u5230 0\u3002 <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c5e09be0ae49504103eb4cb5bc2bff7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} \\cfrac{9x}{4-x^2} = \\cfrac{9(-\\infty)}{-(-\\infty)^2} = \\cfrac{-\\infty}{-(+\\infty)}=\\cfrac{-\\infty}{-\\infty}= \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"374\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6016<\/h3>\n<p>\u6c42\u89e3\u4ee5\u4e0b\u51fd\u6570\u7684\u81f3\u5c11\u65e0\u9650\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-550b7d336f11ad3346cc238a9f5719db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} \\cfrac{-2x^3-3x}{-3x^2+4x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"163\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5206\u5b50\u6bd4\u5206\u6bcd\u5927\u4e00\u7ea7\uff0c\u56e0\u6b64\u4e0d\u5b9a\u5f62\u5f0f \u221e\/\u221e \u7684\u7ed3\u679c\u5c06\u662f\u65e0\u7a77\u5927\u3002\u6b64\u5916\uff0c\u65e0\u7a77\u5927\u7b26\u53f7\u5c06\u4e3a\u8d1f\u6570\uff0c\u56e0\u4e3a\u8d1f\u6570\u4e4b\u95f4\u7684\u6b63\u6570\u4f1a\u8f6c\u5316\u4e3a\u8d1f\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9820c6575934eac4bea0f71a98db09b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty} \\cfrac{-2x^3-3x}{-3x^2+4x-1} = \\cfrac{-2(-\\infty)^3}{-3(-\\infty)^2} =\\cfrac{-2(-\\infty)}{-3(+\\infty)}= \\cfrac{+\\infty}{-\\infty}= \\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"493\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6017<\/h3>\n<p>\u6c42\u89e3\u4ee5\u4e0b\u65e0\u7a77\u8fdc\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5ebe0714beba2eea5d7ab668eb8c75de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\cfrac{2^x-4}{-2x^6+x^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"131\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u6307\u6570\u51fd\u6570\u7684\u9636\u6570\u9ad8\u4e8e\u591a\u9879\u5f0f\u51fd\u6570\uff0c\u56e0\u6b64\u6781\u9650\u5c06\u4e3a\u65e0\u7a77\u5927\u3002\u4f46\u662f\uff0c\u5c06\u6b63\u6570\u9664\u4ee5\u8d1f\u6570\uff0c\u65e0\u7a77\u5927\u7b26\u53f7\u5c06\u4e3a\u8d1f\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7917d9ddbc8ccb39774511497bdefb5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\frac{2^x-4}{-2x^6+x^4}=\\frac{2^{+\\infty}}{-2(+\\infty)^6}=\\frac{+\\infty}{-\\infty}=\\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"350\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6018<\/h3>\n<p>\u4f7f\u7528\u5e73\u65b9\u6839\u8ba1\u7b97\u4ee5\u4e0b\u51fd\u6570\u7684\u65e0\u9650\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d7a236a0525e9580e50ff2e179ca1966_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{\\sqrt{4x^2+1}}{-2x}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"124\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5206\u5b50\u7531\u5e73\u65b9\u6839\u7ec4\u6210\uff0c\u56e0\u6b64\u5176\u6b21\u6570\u4e3a 2\/2=1\u3002\u90a3\u4e48\uff0c\u5206\u5b50\u7684\u6b21\u6570\u7b49\u4e8e\u5206\u6bcd\u7684\u6b21\u6570\uff0c\u56e0\u6b64\u65e0\u7a77\u5927\u4e4b\u95f4\u7684\u65e0\u9650\u4e0d\u786e\u5b9a\u6027\u53ef\u4ee5\u5982\u4e0b\u89e3\u51b3\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ff123dca95b296fce56ef0d4cf80673c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{\\sqrt{4x^2+1}}{-2x}= \\cfrac{\\sqrt{4(+\\infty)^2}}{-2(\\infty)}= \\cfrac{+\\infty}{-\\infty}  = \\cfrac{\\sqrt{4}}{-2}=\\cfrac{2}{-2}=\\bm{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"436\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6019<\/h3>\n<p>\u6c42\u89e3\u4ee5\u4e0b\u5177\u6709\u4e24\u4e2a\u6839\u5f0f\u7684\u51fd\u6570\u7684\u65e0\u9650\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-66a62b591cedd9d53e14613fc16bca97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{\\sqrt[3]{6x^7+2x^3}}{\\sqrt{x^5-3x^4+2x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"173\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5206\u5b50\u7684\u6b21\u6570\u4e3a7\/3=2.33\uff0c\u5206\u6bcd\u7684\u6b21\u6570\u4e3a5\/2=2.5\u3002\u56e0\u6b64\uff0c\u7531\u4e8e\u5206\u5b50\u7684\u6b21\u6570\u5c0f\u4e8e\u5206\u6bcd\u7684\u6b21\u6570\uff0c\u65e0\u7a77\u5927\u4e4b\u95f4\u7684\u4e0d\u786e\u5b9a\u65e0\u9650\u6781\u9650\u4e3a 0\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-681401701d7d7f3fad1879db26659942_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{\\sqrt[3]{6x^7+2x^3}}{\\sqrt{x^5-3x^4+2x}}=\\cfrac{\\sqrt[3]{6(+\\infty)^7}}{\\sqrt{(+\\infty)^5}}=\\cfrac{+\\infty}{+\\infty}=\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"50\" width=\"376\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6020<\/h3>\n<p>\u8ba1\u7b97\u4ee5\u4e0b\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fe6ecfeb0afd1ce82003504bdd2222a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{\\sqrt[5]{x^7-2x^5-1}}{4^{x-2}+3x}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"164\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u65e0\u8bba\u5206\u5b50\u7684\u6b21\u6570\u5982\u4f55\uff0c\u7531\u4e8e\u5206\u6bcd\u4e2d\u6709\u4e00\u4e2a\u6307\u6570\u51fd\u6570\uff0c\u56e0\u6b64\u4e0d\u5b9a\u5f62\u5f0f\u65e0\u7a77\u5927\u7684\u7ed3\u679c\u662f 0\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cc9e15968203ed8d39e04b1f2239b9b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{\\sqrt[5]{x^7-2x^5-1}}{4^{x-2}+3x}=\\cfrac{\\sqrt[5]{(+\\infty)^7}}{4^{+\\infty-2}}=\\cfrac{+\\infty}{+\\infty}=\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"358\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6021<\/h3>\n<p>\u786e\u5b9a\u4ee5\u4e0b\u6709\u7406\u51fd\u6570\u7684\u65e0\u9650\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f0a8401379d90875626b1fbd3714fd01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(\\frac{x^3+1}{x-1}-\\frac{x}{4} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"160\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u9996\u5148\uff0c\u6211\u4eec\u5c1d\u8bd5\u901a\u8fc7\u5c06\u65e0\u7a77\u5927\u4ee3\u5165\u51fd\u6570\u6765\u8ba1\u7b97\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-43df032007d76e00f2f7366e05f9e697_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(\\frac{x^3+1}{x-1}-\\frac{x}{4}\\right)=\\frac{(+\\infty)^3+1}{+\\infty-1}-\\frac{+\\infty}{4} = \\bm{+\\infty -\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"425\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4f46\u6211\u4eec\u53d1\u73b0\u4e0d\u786e\u5b9a\u6027 \u221e \u2013 \u221e\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u5c06\u5206\u6570\u51cf\u5c11\u5230\u4e00\u4e2a\u516c\u5206\u6bcd\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7e2820674bc86d085f6deec7fdf9adf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim\\limits_{x \\to +\\infty} \\left(\\frac{x^3+1}{x-1}-\\frac{x}{4} \\right)=\\\\[5ex]\\displaystyle = \\lim_{x\\to +\\infty}\\left(\\frac{(x^3+1)\\cdot4}{(x-1)\\cdot4}-\\frac{x\\cdot(x-1)}{4\\cdot (x-1)}\\right)=\\\\[5ex]\\displaystyle =\\lim_{x\\to +\\infty}\\left(\\frac{4x^3+4}{4x-4}-\\frac{x^2-x}{4x-4}\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"188\" width=\"302\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u7531\u4e8e\u4e24\u4e2a\u5206\u6570\u73b0\u5728\u5177\u6709\u76f8\u540c\u7684\u5206\u6bcd\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b83\u4eec\u5408\u5e76\u4e3a\u4e00\u4e2a\u5206\u6570\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-93a00027be74b1e60c7ee8537ebe5d9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\left(\\frac{4x^3+4}{4x-4}-\\frac{x^2-x}{4x-4}\\right)=\\lim_{x\\to +\\infty}\\frac{4x^3+4-(x^2-x)}{4x-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"429\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u628a\u5206\u5b50\u653e\u5728\u62ec\u53f7\u91cc\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c7de3ead5b3a5f8bd2ae8d767da693b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty} \\frac{4x^3+4-x^2+x}{4x-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"180\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u786e\u5b9a\u9650\u5236\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7ffb73fbf26fd2b625e43872a9c10ef9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\frac{4x^3+4-x^2+x}{4x-4}=\\frac{4(+\\infty)^3}{4(+\\infty)}=\\frac{+\\infty}{+\\infty} = \\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"384\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u4e0d\u786e\u5b9a\u6027 \u221e\/\u221e \u7ed9\u51fa +\u221e\uff0c\u56e0\u4e3a\u5206\u5b50\u7684\u6b21\u6570\u5927\u4e8e\u5206\u6bcd\u7684\u6b21\u6570\u3002<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6022<\/h3>\n<p>\u5f53 x \u63a5\u8fd1 0 \u65f6\uff0c\u6c42\u89e3\u4ee5\u4e0b\u5206\u6570\u51fd\u6570\u7684\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e783bc22baa422d4b537fae4628fb4a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5}{x^2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"165\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u6211\u4eec\u9996\u5148\u50cf\u5f80\u5e38\u4e00\u6837\u5c1d\u8bd5\u8ba1\u7b97\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-207bc08385f430f0f8c49ac34a10f811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5}{x^2}\\right)=\\frac{-3\\cdot0-2}{0^4}-\\frac{5}{0^2}=\\frac{-2}{0}-\\frac{5}{0}=\\bm{\\infty-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"477\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4f46\u6211\u4eec\u5f97\u5230\u4e86\u4e0d\u5b9a\u5f62\u5f0f \u221e-\u221e\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u5fc5\u987b\u5c06\u51fd\u6570\u7684\u5206\u6570\u51cf\u5c11\u5230\u4e00\u4e2a\u516c\u5206\u6bcd\u3002<\/p>\n<p class=\"has-text-align-left\">\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c x <sup>4<\/sup>\u662f x <sup>2<\/sup>\u7684\u500d\u6570\uff0c\u56e0\u6b64\u53ea\u9700\u5c06\u7b2c\u4e8c\u4e2a\u5206\u6570\u7684\u5206\u5b50\u548c\u5206\u6bcd\u4e58\u4ee5 x <sup>2<\/sup> \uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u786e\u4fdd\u4e24\u4e2a\u5206\u6570\u5177\u6709\u76f8\u540c\u7684\u5206\u6bcd\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-876115dc1fb49e81373d70be5fdcfb5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x \\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5}{x^2}\\right)=\\\\[5ex]\\displaystyle =\\lim_{x \\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5\\cdot x^2}{x^2\\cdot x^2} \\right)=\\\\[5ex]\\displaystyle =\\lim_{x\\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5x^2}{x^4}\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"186\" width=\"235\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u73b0\u5728\u53ef\u4ee5\u51cf\u53bb\u4e24\u4e2a\u5206\u6570\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bf56e81e075d9ac498e9df87a94a675f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5x^2}{x^4}\\right)=\\lim_{x\\to 0}\\frac{-3x-2-5x^2 }{x^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"346\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u518d\u6b21\u5c1d\u8bd5\u89e3\u51b3\u9650\u5236\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b231cf80ccb03d1287c1aab47769bc34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 0}  \\cfrac{-3x-2-5x^2 }{x^4} =\\cfrac{-3\\cdot 0-2-5\\cdot 0^2}{0^4}=\\frac{-2}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"370\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4f46\u6211\u4eec\u6700\u7ec8\u5f97\u5230\u4e86\u4ece\u96f6\u5f00\u59cb\u7684\u5e38\u6570\u7684\u4e0d\u786e\u5b9a\u6027\u3002\u56e0\u6b64\u6709\u5fc5\u8981\u8ba1\u7b97\u51fd\u6570\u7684\u6a2a\u5411\u6781\u9650\u3002 <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c4ced459b1e0da92f03d9d9515b6ea68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 0^{-}} \\frac{-3x-2-5x^2}{x^4}=\\frac{-2}{+0}=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"262\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-239f065e0fe7bb4055e63a8477c030f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 0^{+}}\\frac{-3x-2-5x^2}{x^4}=\\frac{-2}{+0}=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"262\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u603b\u4e4b\uff0c\u7531\u4e8e\u51fd\u6570\u5728\u70b9 x=0 \u5904\u7684\u4e24\u4e2a\u6a2a\u5411\u6781\u9650\u4e3a -\u221e\uff0c\u56e0\u6b64\u8be5\u6781\u9650\u7684\u89e3\u4e3a -\u221e\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-30ab5fa39e1b25568d55de0cc4267dc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 0^-}f(x)=\\lim_{x \\to 0^+}f(x)=-\\infty\\ \\longrightarrow \\  \\lim_{x \\to 0}f(x)= \\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"401\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6023<\/h3>\n<p>\u6c42\u89e3\u4ee5\u4e0b\u6709\u6839\u51fd\u6570\u7684\u65e0\u9650\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5fb2be8c217ffddadf1b3d9d55f100c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(4x^2-\\sqrt{x^4+1}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"182\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5c1d\u8bd5\u6c42\u89e3\u6781\u9650\uff0c\u6211\u4eec\u5f97\u5230\u4e0d\u786e\u5b9a\u6027\u65e0\u7a77\u5927\u51cf\u53bb\u65e0\u7a77\u5927\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a1a6b3ff08a703378b8cfb1b5e6532c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(4x^2-\\sqrt{x^4+1}\\right)=4(+\\infty)^2-\\sqrt{(+\\infty)^4}=\\bm{+\\infty -\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"456\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u7531\u4e8e\u51fd\u6570\u4e2d\u6709\u6839\u5f0f\uff0c\u6240\u4ee5\u5fc5\u987b\u4e58\u4ee5\u548c\u9664\u4ee5\u5171\u8f6d\u6839\u5f0f\u8868\u8fbe\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3c4cdc9585a792800b8c903745ecc7c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\left(4x^2-\\sqrt{x^4+1} \\right)=\\lim_{x \\to +\\infty}\\frac{\\left(4x^2-\\sqrt{x^4+1}\\right)\\cdot\\left(4x^2+\\sqrt{x^4+1}\\right)}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"538\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5728\u5206\u5b50\u4e2d\uff0c\u6211\u4eec\u6709\u4e00\u4e2a\u548c\u4e0e\u5dee\u7684\u663e\u7740\u4e58\u79ef\uff0c\u5b83\u7b49\u4e8e\u5e73\u65b9\u5dee\u3002\u7136\u800c\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1aab32f1a28189a4ce96f3816f11a02e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\left(4x^2\\right)^2-\\left(\\sqrt{x^4+1}\\right)^2}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"216\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u5c06\u6839\u5f0f\u7b80\u5316\u4e3a\u5e73\u65b9\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6d86adc198c4fb2cd1d99c94e5b8430e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\bigl(4x^2\\bigr)^2-(x^4+1)}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"186\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u5bf9\u5206\u5b50\u8fdb\u884c\u8fd0\u7b97\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c07c403048b4d3e40a8034333ff069c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{16x^4-x^4-1}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"163\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c138b064a8fa3142cb2d50782807ebb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{15x^4-1}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"163\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\u6211\u4eec\u627e\u5230\u4e86\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cdb8845be6c640f0370961c3a52598d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{15x^4-1}{4x^2+\\sqrt{x^4+1}}=\\frac{15(+\\infty)^4}{4(+\\infty)^2+\\sqrt{(+\\infty)^4}}=\\frac{+\\infty}{+\\infty}= \\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"460\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u4e0d\u786e\u5b9a\u6027\u65e0\u7a77\u5927\u9664\u4ee5\u65e0\u7a77\u5927\u53d8\u5f97\u66f4\u52a0\u65e0\u9650\uff0c\u56e0\u4e3a\u5206\u5b50\u7684\u6b21\u6570\u5927\u4e8e\u5206\u6bcd\u7684\u6b21\u6570\uff08\u56de\u60f3\u4e00\u4e0b\uff0c\u5e73\u65b9\u6839\u5c06\u6b21\u6570\u51cf\u5c11\u4e86\u4e8c\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ded55d5413ed7bccc29e8228df205f19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{x^4} = x^{4\/2} = x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -1px;\"><\/p>\n<p> \uff09\u3002<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e6024<\/h3>\n<p>\u5f53 x \u63a5\u8fd1\u65e0\u7a77\u5927\u65f6\uff0c\u6c42\u89e3\u4ee5\u4e0b\u65e0\u7406\u51fd\u6570\u7684\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7d9f21f0159778cdb1f0710e1a9e0023_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(2x-1-\\sqrt{4x^2+1}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"214\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u9996\u5148\uff0c\u6211\u4eec\u5c1d\u8bd5\u7167\u5e38\u8ba1\u7b97\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5419056e772f9d11884cae7e315ca947_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(2x-1-\\sqrt{4x^2+1}\\right)=2(+\\infty)-\\sqrt{4(+\\infty)^2}=\\bm{+\\infty -\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"489\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4f46\u8fd9\u5bfc\u81f4\u65e0\u7a77\u5927\u4e4b\u5dee\u7684\u4e0d\u786e\u5b9a\u6027\u3002\u56e0\u6b64\uff0c\u7531\u4e8e\u51fd\u6570\u6709\u6839\uff0c\u6211\u4eec\u5fc5\u987b\u5c06\u8868\u8fbe\u5f0f\u4e58\u4ee5\u548c\u9664\u4ee5\u5171\u8f6d\u6839\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bde8a1f86cf7be80170b9595b5a822df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\left(2x-1-\\sqrt{4x^2+1}\\right)\\cdot\\left(2x-1+\\sqrt{4x^2+1}\\right)}{2x-1 +\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"393\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u5c06\u5206\u6570\u5206\u5b50\u7684\u663e\u7740\u76f8\u7b49\u6027\u5206\u7ec4\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e074e8c7841e0951ae03d6dfd2bfd1b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\left(2x-1\\right)^2-\\left(\\sqrt{4x^2+1}\\right)^2}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"245\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u6c42\u89e3\u5e73\u65b9\u6839\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-17beafd120a7fc185e1499671fb4421a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\left(2x-1\\right)^2-\\left(4x^2+1\\right)}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"218\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u6c42\u89e3\u5dee\u5e73\u65b9\u7684\u663e\u7740\u6052\u7b49\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4a34cb3941c92a785c11c50ecaa1e438_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{4x^2+1-4x-\\left(4x^2+1\\right)}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"245\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u5bf9\u5206\u5b50\u8fdb\u884c\u8fd0\u7b97\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1a2e8d86f22087e775650d36bf78e719_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{4x^2+1-4x-4x^2-1}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"228\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2f25890bccb1eaa4c7aa7338f3a25f6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\frac{-4x}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"195\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u8ba1\u7b97\u65e0\u7a77\u5927\u7684\u6781\u9650\u503c\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6986ded778a6220e3ad9d6c6bf873451_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty} \\cfrac{-4x }{2x-1 +\\sqrt{4x^2+1} } = \\cfrac{-4(+\\infty) }{2(+\\infty)+\\sqrt{4(+\\infty)^2} } = \\cfrac{-\\infty}{+\\infty} =\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"458\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5c3d\u7ba1\u5206\u6bcd\u4e2d\u6709\u4e00\u4e2a x \u5e73\u65b9\uff0c\u4f46\u5b83\u7684\u6b21\u6570\u5b9e\u9645\u4e0a\u662f 1\uff0c\u56e0\u4e3a\u5b83\u4f4d\u4e8e\u6839\u5185\u90e8\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0decc88d206f476d332becb025b8eeaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4x^2} =\\sqrt{4}\\cdot \\sqrt{x^2} = \\sqrt{4}\\cdot x^{2\/2} =\\sqrt{4} x^1=\\sqrt{4}x .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"351\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u4e0d\u786e\u5b9a\u6027 -\u221e\/+\u221e \u7684\u7ed3\u679c\u662f\u6700\u9ad8\u6b21 x \u7684\u7cfb\u6570\u76f8\u9664\uff0c\u56e0\u4e3a\u5206\u5b50\u7684\u6b21\u6570\u4e0e\u5206\u6bcd\u7684\u6b21\u6570\u76f8\u540c\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8eb19af7ca51c14245db81bd6781b881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{-4x}{2x-1+\\sqrt{4x^2+1} }=\\frac{-\\infty}{+\\infty}=\\frac{-4}{2+\\sqrt{4}}=\\frac{-4}{2+2}=\\frac{-4}{4}=\\bm{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"499\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u5206\u6bcd\u4e2d\u6709\u4e24\u4e2a\u4e00\u7ea7\u9879<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c973910499b6b5a4828e213dc33f948d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl(2x\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -7px;\"><\/p>\n<p>\u548c<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c623cb17f27418239e3fcf7c2ec09946_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4x^2}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"46\" style=\"vertical-align: -7px;\"><\/p>\n<p>\uff0c\u4e3a\u4e86\u89e3\u51b3\u4e0d\u786e\u5b9a\u6027-\u221e\/+\u221e\uff0c\u9700\u8981\u53d6\u4e00\u6b21\u9879\u7684\u6240\u6709\u7cfb\u6570\uff0c\u4e5f\u5c31\u662f\u8bf4<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u7684<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-da4556c0a02b580047678d308649edf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u548c<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-65ddaa07508d3929b6969a5e4e6baddf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\"><\/p>\n<p>\u7684<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e8a851efdbfbb4531c82837d5a61edd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4x^2}.\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -1px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e60 25<\/h3>\n<p>\u8ba1\u7b97\u5f53 x \u63a5\u8fd1\u4ee5\u4e0b\u5e26\u6709\u5206\u6570\u7684\u51fd\u6570\u7684 1 \u65f6\u7684\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-480bb119c1303a7afa394d812b0e7602_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1}\\left(\\frac{1}{1-x}-\\frac{3}{1-x^3}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"173\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u901a\u8fc7\u5c1d\u8bd5\u6c42\u6781\u9650\uff0c\u6211\u4eec\u5f97\u5230\u65e0\u7a77\u5927\u51cf\u53bb\u65e0\u7a77\u5927\u7684\u4e0d\u786e\u5b9a\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d11d45ea6681f3645773f6e0df8cce9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1}\\left(\\frac{1}{1-x}-\\frac{3}{1-x^3}\\right)=\\frac{1}{1-1}--\\frac{3}{1-1^3}=\\frac{1}{0}-\\frac{3}{0}=\\bm{\\infty-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"480\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u6211\u4eec\u5fc5\u987b\u5c06\u5206\u6570\u5316\u7b80\u4e3a\u4e00\u4e2a\u516c\u5206\u6bcd\uff0c\u6216\u8005\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u5fc5\u987b\u5c06\u4e00\u4e2a\u5206\u6570\u7684\u5206\u5b50\u548c\u5206\u6bcd\u4e58\u4ee5\u53e6\u4e00\u4e2a\u5206\u6570\u7684\u5206\u6bcd\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-75bf3ffa177f32711c5509ce5fe5992d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x \\to 1}\\left(\\frac{1}{1-x}-\\frac{3}{1-x^3} \\right)=\\\\[5ex]\\displaystyle =\\lim_{x\\to 1}\\left( \\frac{1\\cdot(1-x^3)}{(1-x)\\cdot(1-x^3)}-\\frac{3\\cdot(1-x)}{(1-x^3)\\cdot(1-x)}\\right)=\\\\[5ex]\\displaystyle =\\lim_{x \\to 1}\\left(\\frac{1-x^3}{1-x-x^3+x^4}-\\frac{3-3x}{1-x-x^3+x^4}\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"186\" width=\"382\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u7531\u4e8e\u73b0\u5728\u4e24\u4e2a\u5206\u6570\u5177\u6709\u76f8\u540c\u7684\u5206\u6bcd\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b83\u4eec\u653e\u5728\u4e00\u8d77\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c381a263e89e5a60ff0e6df9367a8ab1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\left(\\frac{1-x^3}{1-x-x^3+x^4}-\\frac{3-3x}{1-x-x^3+x^4}\\right)=\\lim_{x\\to 1}\\frac{1-x^3-(3-3x)}{1-x-x^3+x^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"517\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u7ecf\u8425\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-05279cd25d55f5c50edfb5f82929701b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1} \\cfrac{1-x^3-3+3x}{1-x-x^3+x^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-818107141eb339d788408e23078ddda9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1} \\cfrac{-x^3+3x-2}{x^4-x^3-x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u518d\u6b21\u5c1d\u8bd5\u89e3\u51b3\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9d0a31b51faff7e77e778fba66fdbaa1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\frac{-x^3+3x-2}{x^4-x^3-x+1}=\\frac{-1^3+3\\cdot1-2}{1^4-1^3-1+1}=\\mathbf{\\frac{0}{0}}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"335\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4f46\u6211\u4eec\u53d1\u73b0\u4e0d\u786e\u5b9a\u6027\u4e3a\u96f6\u9664\u4ee5\u96f6\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u5fc5\u987b\u5bf9\u5206\u5b50\u548c\u5206\u6bcd\u7684\u591a\u9879\u5f0f\u8fdb\u884c\u56e0\u5f0f\u5206\u89e3\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e5b8321a511b5e370abe8844bf9624ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\frac{-x^3+3x-2}{x^4-x^3-x+1}=\\lim_{x \\to 1}\\frac{-(x-1)^2(x+2)}{(x-1)^2(x^2+x+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"369\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u6211\u4eec\u901a\u8fc7\u5220\u9664\u5206\u5b50\u548c\u5206\u6bcd\u4e2d\u91cd\u590d\u7684\u56e0\u5b50\u6765\u7b80\u5316\u5206\u6570\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ab5629bd2fabeb755da37d3abea335b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\frac{-\\cancel{(x-1)^2}(x+2)}{\\cancel{(x-1)^2}(x^2+x+1)}=\\lim_{x \\to 1}\\frac{-(x+2)}{x^2+x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"329\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u89e3\u51b3\u4e86\u9650\u5236\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dbb1676133fe1e33fb4d18078b945959_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\frac{-(x+2)}{x^2+x+1}=\\frac{-(1+2)}{1^2+1+1}=\\frac{-3}{3}=\\bm{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"316\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u5728\u8fd9\u91cc\uff0c\u60a8\u5c06\u4e86\u89e3\u5982\u4f55\u6c42\u89e3\u6240\u6709\u7c7b\u578b\u7684\u65e0\u7a77\u5927\u6781\u9650\uff1a\u591a\u9879\u5f0f\u3001\u6709\u7406\u51fd\u6570\u3001\u6307\u6570\u51fd\u6570\u3001\u6709\u6839\u51fd\u6570\u3001\u65e0\u7a77\u5927\u4e0d\u786e\u5b9a\u6027&#038;#8230 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/cn\/%e6%97%a0%e7%a9%b7%e5%a4%a7%e7%9a%84%e6%9e%81%e9%99%90\/\"> <span class=\"screen-reader-text\">\u65e0\u7a77\u5927\u7684\u6781\u9650<\/span> \u67e5\u770b\u5168\u6587 &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[11],"tags":[],"class_list":["post-21","post","type-post","status-publish","format-standard","hentry","category-11"],"yoast_head":"<!-- This site is optimized 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