{"id":365,"date":"2023-07-04T04:08:40","date_gmt":"2023-07-04T04:08:40","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e4%b8%8d%e7%a1%ae%e5%ae%9a%e6%80%a7-%e6%97%a0%e9%99%90%e5%87%8f%e6%97%a0%e9%99%90-%e2%88%9e-%e2%88%9e\/"},"modified":"2023-07-04T04:08:40","modified_gmt":"2023-07-04T04:08:40","slug":"%e4%b8%8d%e7%a1%ae%e5%ae%9a%e6%80%a7-%e6%97%a0%e9%99%90%e5%87%8f%e6%97%a0%e9%99%90-%e2%88%9e-%e2%88%9e","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e4%b8%8d%e7%a1%ae%e5%ae%9a%e6%80%a7-%e6%97%a0%e9%99%90%e5%87%8f%e6%97%a0%e9%99%90-%e2%88%9e-%e2%88%9e\/","title":{"rendered":"\u4e0d\u786e\u5b9a\u6027 \u65e0\u7a77\u51cf\u65e0\u7a77 (\u65e0\u7a77\u5927-\u65e0\u7a77\u5927)"},"content":{"rendered":"<p>\u5728\u672c\u6587\u4e2d\uff0c\u6211\u4eec\u5c06\u89e3\u91ca\u5982\u4f55\u89e3\u51b3\u65e0\u7a77\u8d1f\u65e0\u7a77 (\u221e-\u221e) \u4e0d\u786e\u5b9a\u6027\u3002\u60a8\u5c06\u627e\u5230\u5177\u6709\u4e0d\u540c\u7c7b\u578b\u51fd\u6570\u7684\u8fd9\u79cd\u4e0d\u786e\u5b9a\u6027\u7684\u793a\u4f8b\uff0c\u6b64\u5916\uff0c\u60a8\u5c06\u80fd\u591f\u901a\u8fc7\u9010\u6b65\u89e3\u51b3\u4e0d\u786e\u5b9a\u6027\u65e0\u9650\u51cf\u65e0\u7a77\u7684\u7ec3\u4e60\u8fdb\u884c\u7ec3\u4e60\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"resolver-la-indeterminacion-infinito-menos-infinito\"><\/span>\u89e3\u51b3\u4e0d\u786e\u5b9a\u6027\u65e0\u7a77\u51cf\u65e0\u7a77<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5f53\u51fd\u6570\u7684\u6781\u9650\u7ed9\u51fa\u65e0\u7a77\u5927\u51cf\u65e0\u7a77\u5927\u65f6\uff0c\u5c31\u610f\u5473\u7740\u5b83\u662f\u4e0d\u5b9a\u6027\uff08\u6216\u4e0d\u5b9a\u5f62\u5f0f\uff09\u3002\u4e5f\u5c31\u662f\u8bf4<strong>\uff0c\u7ed9\u51fa\u4e0d\u786e\u5b9a\u6027\u51cf\u53bb\u65e0\u7a77\u5927\u7684\u51fd\u6570\u7684\u6781\u9650<\/strong>\u4e0d\u80fd\u901a\u8fc7\u6267\u884c\u76f4\u63a5\u8ba1\u7b97\u6765\u786e\u5b9a\uff0c\u800c\u662f\u5fc5\u987b\u6267\u884c\u521d\u6b65\u7a0b\u5e8f\u3002<\/p>\n<p>\u56e0\u6b64\uff0c\u8981<strong>\u89e3\u51b3\u65e0\u7a77\u51cf\u65e0\u7a77\u7684\u4e0d\u786e\u5b9a\u6027\uff0c<\/strong>\u6211\u4eec\u5fc5\u987b\u9996\u5148\u5e94\u7528\u53d6\u51b3\u4e8e\u51fd\u6570\u7c7b\u578b\u7684\u7a0b\u5e8f\uff1a\u5982\u679c\u662f\u591a\u9879\u5f0f\u51fd\u6570\uff0c\u5219\u53ef\u4ee5\u901a\u8fc7\u6bd4\u8f83\u6765\u8ba1\u7b97\uff0c\u5982\u679c\u662f\u6709\u7406\u51fd\u6570\uff0c\u5219\u5fc5\u987b\u5c06\u5206\u6570\u7b80\u5316\u4e3a\u516c\u5206\u6bcd\uff0c\u5982\u679c\u662f\u65e0\u7406\u51fd\u6570\uff0c\u5219\u5fc5\u987b\u4e58\u4ee5\u5171\u8f6d\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-b1bfc56d2079c86e8ad6e1943311b730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to+\\infty}\\Bigl(f(x)-g(x)\\Bigr)=\\infty-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"236\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u901a\u8fc7\u793a\u4f8b\u4e86\u89e3\u5728\u6bcf\u79cd\u7c7b\u578b\u7684\u51fd\u6570\u4e2d\u5982\u4f55\u89e3\u51b3\u4e0d\u786e\u5b9a\u6027\u65e0\u7a77\u5927\u51cf\u53bb\u65e0\u7a77\u5927\u7684\u95ee\u9898\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-infinito-menos-infinito-en-funciones-polinomicas\"><\/span>\u591a\u9879\u5f0f\u51fd\u6570\u4e2d\u7684\u65e0\u7a77\u51cf\u65e0\u7a77\u4e0d\u786e\u5b9a\u6027<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\">\u5728\u591a\u9879\u5f0f\u4e2d\uff0c\u4e0d\u786e\u5b9a\u6027\u65e0\u7a77\u5927\u51cf\u53bb\u65e0\u7a77\u5927\u7b49\u4e8e\u6700\u9ad8\u9636\u65e0\u7a77\u5927\uff0c\u5373\u6700\u9ad8\u9636\u9879\u51b3\u5b9a\u65e0\u7a77\u5927\u7684\u6b63\u53f7\u6216\u8d1f\u53f7\u3002<\/p>\n<\/div>\n<p>\u4f8b\u5982\uff0c\u770b\u770b\u4ee5\u4e0b\u591a\u9879\u5f0f\u51fd\u6570\u7684\u6781\u9650\uff0c\u5b83\u7ed9\u51fa\u4e86\u4e0d\u786e\u5b9a\u5f62\u5f0f\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-f61115731d29dc9f05941968417c9443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to+\\infty}\\bigl(x^2-3x\\bigr)=(+\\infty)^2-3\\cdot (\\infty)=+\\infty-\\infty=+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"421\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p>\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u9879 x <sup>2<\/sup>\u662f\u4e8c\u6b21\u9879\uff0c\u9879 3x \u662f\u4e00\u6b21\u9879\uff0c\u56e0\u6b64\u5355\u9879\u5f0f x <sup>2<\/sup>\u5360\u4e3b\u5bfc\u5730\u4f4d\uff0c\u56e0\u4e3a\u5b83\u662f\u66f4\u9ad8\u9636\u7684\u3002\u56e0\u6b64\uff0c\u6781\u9650\u7684\u7ed3\u679c\u662f\u4ece\u8fd9\u9879\u5f97\u5230\u7684\u65e0\u7a77\u5927\u3002<\/p>\n<p>\u770b\u770b\u8fd9\u4e9b\u5176\u4ed6\u4f8b\u5b50\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-3bfb5cd294a19de382f74738af6be724_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to+\\infty}\\bigl(x^5-4x^2-3x\\bigr)=(+\\infty)^5=+\\infty\\\\[5ex]\\displaystyle\\lim_{x\\to-\\infty}\\bigl(-3x^2-5x\\bigr)=-3\\cdot (-\\infty)^2=-3\\cdot \\infty=-\\infty\\\\[5ex]\\displaystyle\\lim_{x\\to+\\infty}\\bigl(x^7-5x^4+x^3-2x-10\\bigr)=(+\\infty)^7=+\\infty\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"150\" width=\"387\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u7b80\u800c\u8a00\u4e4b\uff0c\u5f53\u6211\u4eec\u5bf9\u591a\u9879\u5f0f\u51fd\u6570 \u4e2d\u7684\u65e0\u7a77\u5927\u8fdb\u884c\u9650\u5236\u65f6<strong>\uff0c\u6211\u4eec\u5fc5\u987b\u7b80\u5355\u5730\u5c06\u65e0\u7a77\u5927\u4ee3\u5165\u6700\u9ad8\u6b21\u9879<\/strong>\uff0c\u800c\u5ffd\u7565\u6240\u6709\u5176\u4ed6\u9879\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-infinito-menos-infinito-con-fracciones\"><\/span>\u4e0d\u786e\u5b9a\u6027 \u65e0\u9650\u51cf\u65e0\u7a77 \u5e26\u5206\u6570<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\"><strong>\u5f53\u4ee3\u6570\u5206\u6570\u7684\u52a0\u6cd5\u6216\u51cf\u6cd5\u4e2d\u51fa\u73b0\u65e0\u7a77\u5927\u51cf\u65e0\u7a77\u5927\u7684\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\u8ba1\u7b97\u5e26\u5206\u6570\u51fd\u6570\u4e2d\u7684\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\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>\u6240\u4ee5\u9996\u5148\u6211\u4eec\u9700\u8981\u8fdb\u884c\u5206\u6570\u51cf\u6cd5\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\u7528\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\uff0c\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<p> <span style=\"color:#ff951b\">\u27a4<\/span><strong>\u770b\uff1a<\/strong> <a href=\"https:\/\/mathority.org\/cn\/\u65e0\u9650-%e2%88%9e-%e2%88%9e-\u4e4b\u95f4\u7684\u4e0d\u786e\u5b9a\u6027\/\"><span style=\"text-decoration: underline;\">\u65e0\u7a77\u5927\u4e4b\u95f4\u7684\u65e0\u7a77\u5927\u662f\u4ec0\u4e48\uff1f<\/span><\/a> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-infinito-menos-infinito-con-raices\"><\/span>\u4e0d\u786e\u5b9a\u6027 \u65e0\u9650\u51cf\u65e0\u7a77 \u6709\u6839<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\">\u5f53<strong>\u6839\u5f0f\u52a0\u6cd5\u6216\u6839\u5f0f\u51cf\u6cd5\u4e2d\u51fa\u73b0\u65e0\u7a77\u5927\u51cf\u65e0\u7a77\u5927\u7684\u4e0d\u786e\u5b9a\u6027<\/strong>\u65f6\uff0c\u5fc5\u987b\u5148\u5c06\u51fd\u6570\u4e58\u4ee5\u548c\u9664\u4ee5\u5171\u8f6d\u6839\u5f0f\u8868\u8fbe\u5f0f\uff0c\u7136\u540e\u6c42\u89e3\u6781\u9650\u3002<\/p>\n<\/div>\n<p>\u6211\u4eec\u5c06\u901a\u8fc7\u9010\u6b65\u793a\u4f8b\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\u89e3\u51b3\u5e26\u6709\u6839\u5f0f\u7684\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\u6211\u4eec\u9700\u8981\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\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-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-la-indeterminacion-infinito-menos-infinito\"><\/span>\u89e3\u51b3\u4e86\u65e0\u7a77\u8d1f\u65e0\u7a77\u4e0d\u786e\u5b9a\u6027\u95ee\u9898<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e601<\/h3>\n<p>\u5f53 x \u63a5\u8fd1\u65e0\u7a77\u5927\u65f6\uff0c\u6c42\u89e3\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-bac36c53d4e34c6e9972009b34a64c21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}(7x^2-2x^3)\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"134\" 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\u8fd9\u4e2a\u6781\u9650\u4e2d\uff0c\u6700\u9ad8\u9636\u9879\u662f\u4e09\u6b21\uff0c\u6240\u4ee5\u6211\u4eec\u5173\u6ce8\u4ece\u8fd9\u9879\u5f97\u5230\u7684\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-eee2b50312f5fd4225c85387c311eec5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}(+7x^2-2x^3)=+\\infty^2-\\infty^3=+\\infty-\\infty=\\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"408\" 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\u4e602<\/h3>\n<p>\u5f53 x \u63a5\u8fd1\u8d1f\u65e0\u7a77\u5927\u65f6\uff0c\u8ba1\u7b97\u4ee5\u4e0b\u591a\u9879\u5f0f\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-12b903d03d726c28b625fe3f5ba4b3c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -\\infty}(-5x^3-9x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"148\" 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\">\u8d1f\u65e0\u7a77\u5927\u7684\u7acb\u65b9\u4ecd\u4e3a\u8d1f\uff0c\u4f46\u5e73\u65b9\u540e\u53d8\u4e3a\u6b63\u3002\u540e\u6765\u867d\u7136\u4ed6\u4eec\u7684\u7b26\u53f7\u88ab\u524d\u9762\u7684\u7cfb\u6570\u4fee\u6539\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-172f2927f65e61079b13abd02234f1c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to -\\infty}(-5x^3+x^2)=\\\\[3ex]=-5(-\\infty)^3-9(-\\infty)^2=\\\\[3ex]=-5\\cdot (-\\infty)-9\\cdot \\infty=\\\\[3ex]=+\\infty-\\infty\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"153\" width=\"196\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u7136\u540e\uff0c\u4e0d\u5b9a\u5f62\u5f0f\u65e0\u7a77\u5927\u51cf\u65e0\u7a77\u5927\u7531\u6700\u9ad8\u9636\u9879 (-5x <sup>3<\/sup> ) \u5b9a\u4e49\uff0c\u4ece\u4e2d\u6211\u4eec\u83b7\u5f97\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-dfd806b31a588234442f48fa5ae8b751_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -\\infty}(-5x^3+x^2)=\\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"194\" 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\u4e603<\/h3>\n<p>\u786e\u5b9a\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-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\u66ff\u6362\u51fd\u6570\u4e2d\u7684\u65e0\u7a77\u5927\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\u6700\u7ec8\u5f97\u5230\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\u4e604<\/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\u9700\u8981\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>\u6211\u4eec\u5c06\u5f97\u5230\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\u5e38\u6570\u9664\u4ee5\u96f6\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\u4e605<\/h3>\n<p>\u6c42\u89e3\u4ee5\u4e0b\u6709\u6839\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-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\u6211\u4eec\u9700\u8981\u5c06\u5176\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\u7528\u5206\u5b50\u6765\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\uff0c\u6211\u4eec\u627e\u5230\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\u65e0\u9650\u4e0d\u786e\u5b9a\u6027\u9664\u4ee5\u65e0\u7a77\u5927\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\u4e606<\/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\u5b83\u6700\u7ec8\u7ed9\u6211\u4eec\u5e26\u6765\u4e86\u65e0\u7a77\u5927\u4e4b\u5dee\u7684\u4e0d\u786e\u5b9a\u6027\u3002\u56e0\u6b64\uff0c\u7531\u4e8e\u51fd\u6570\u6709\u6839\uff0c\u6211\u4eec\u9700\u8981\u5c06\u8868\u8fbe\u5f0f\u4e0e\u5171\u8f6d\u6839\u5f0f\u76f8\u4e58\u548c\u76f8\u9664\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\u7528\u5206\u5b50\u6765\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\u7531\u4e8e\u5206\u5b50\u7684\u6b21\u6570\u4e0e\u5206\u6bcd\u7684\u6b21\u6570\u76f8\u540c\uff0c\u56e0\u6b64\u4e0d\u786e\u5b9a\u6027-\u221e\/+\u221e\u7684\u7ed3\u679c\u662f\u8f83\u9ad8\u6b21\u6570x\u7684\u7cfb\u6570\u7684\u9664\u6cd5\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\u4e607<\/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\">\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\u8fd9\u4e24\u4e2a\u5206\u6570\u73b0\u5728\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\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\u672c\u6587\u4e2d\uff0c\u6211\u4eec\u5c06\u89e3\u91ca\u5982\u4f55\u89e3\u51b3\u65e0\u7a77\u8d1f\u65e0\u7a77 (\u221e-\u221e) \u4e0d\u786e\u5b9a\u6027\u3002\u60a8\u5c06\u627e\u5230\u5177\u6709\u4e0d\u540c\u7c7b\u578b\u51fd\u6570\u7684\u8fd9\u79cd\u4e0d\u786e\u5b9a\u6027\u7684\u793a\u4f8b\uff0c\u6b64 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/cn\/%e4%b8%8d%e7%a1%ae%e5%ae%9a%e6%80%a7-%e6%97%a0%e9%99%90%e5%87%8f%e6%97%a0%e9%99%90-%e2%88%9e-%e2%88%9e\/\"> <span class=\"screen-reader-text\">\u4e0d\u786e\u5b9a\u6027 \u65e0\u7a77\u51cf\u65e0\u7a77 (\u65e0\u7a77\u5927-\u65e0\u7a77\u5927)<\/span> \u67e5\u770b\u5168\u6587 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