{"id":20,"date":"2023-09-17T11:08:12","date_gmt":"2023-09-17T11:08:12","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e4%b8%8d%e7%a1%ae%e5%ae%9a%e7%b1%bb%e5%9e%8b%e4%b8%8d%e7%a1%ae%e5%ae%9a%e6%9e%81%e9%99%90\/"},"modified":"2023-09-17T11:08:12","modified_gmt":"2023-09-17T11:08:12","slug":"%e4%b8%8d%e7%a1%ae%e5%ae%9a%e7%b1%bb%e5%9e%8b%e4%b8%8d%e7%a1%ae%e5%ae%9a%e6%9e%81%e9%99%90","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e4%b8%8d%e7%a1%ae%e5%ae%9a%e7%b1%bb%e5%9e%8b%e4%b8%8d%e7%a1%ae%e5%ae%9a%e6%9e%81%e9%99%90\/","title":{"rendered":"\u4e0d\u786e\u5b9a\u7684\u7c7b\u578b\uff08\u4e0d\u786e\u5b9a\u6781\u9650\uff09"},"content":{"rendered":"<p>\u5728\u8fd9\u7bc7\u6587\u7ae0\u4e2d\uff0c\u6211\u4eec\u89e3\u91ca\u4ec0\u4e48\u662f\u4e0d\u786e\u5b9a\u6027\u3002\u60a8\u5c06\u53d1\u73b0\u6240\u6709\u7c7b\u578b\u7684\u4e0d\u786e\u5b9a\u6027\u662f\u4ec0\u4e48\u4ee5\u53ca\u5982\u4f55\u89e3\u51b3\u5b83\u4eec\u3002\u6b64\u5916\uff0c\u60a8\u5c06\u80fd\u591f\u770b\u5230\u9010\u6b65\u89e3\u51b3\u6240\u6709\u4e0d\u786e\u5b9a\u6027\u51fd\u6570\u6781\u9650\u7684\u7ec3\u4e60\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-las-indeterminaciones\"><\/span>\u4ec0\u4e48\u662f\u4e0d\u786e\u5b9a\u6027\uff1f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>\u4e0d\u5b9a\u5f0f\uff0c\u4e5f\u79f0\u4e3a\u4e0d\u5b9a\u5f62\u5f0f\uff0c\u662f\u51fa\u73b0\u5728\u7ed3\u679c\u672a\u5b9a\u4e49\u7684\u51fd\u6570\u6781\u9650\u8ba1\u7b97\u4e2d\u7684\u6570\u5b66\u8868\u8fbe\u5f0f\u3002<\/strong>\u56e0\u6b64\uff0c\u4e3a\u4e86\u89e3\u51b3\u6781\u9650\u7684\u4e0d\u786e\u5b9a\u6027\uff0c\u6709\u5fc5\u8981\u5e94\u7528\u53d6\u51b3\u4e8e\u51fd\u6570\u7c7b\u578b\u7684\u521d\u6b65\u7a0b\u5e8f\u3002<\/p>\n<p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5f53\u6211\u4eec\u5f97\u5230\u4e0d\u786e\u5b9a\u6027\u65f6\uff0c\u5e76\u4e0d\u610f\u5473\u7740\u6781\u9650\u4e0d\u5b58\u5728\u6216\u65e0\u6cd5\u6c42\u89e3\uff0c\u800c\u662f\u610f\u5473\u7740\u6211\u4eec\u5fc5\u987b\u5bf9\u51fd\u6570\u8fdb\u884c\u66f4\u6539\u624d\u80fd\u627e\u5230\u6781\u9650\u7684\u89e3\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"tipos-de-indeterminaciones\"><\/span>\u4e0d\u786e\u5b9a\u6027\u7684\u7c7b\u578b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u4e0d\u786e\u5b9a\u6216\u4e0d\u786e\u5b9a\u5f62\u5f0f\u5206\u4e3a\u4ee5\u4e0b\u7c7b\u578b\uff1a<\/p>\n<ul>\n<li><strong>\u4e0d\u786e\u5b9a\u6027 \u65e0\u7a77\u5927\u51cf\u65e0\u7a77\u5927<\/strong>(\u65e0\u7a77\u5927-\u65e0\u7a77\u5927)<\/li>\n<li><strong>\u96f6\u4e4b\u95f4\u7684\u4e0d\u786e\u5b9a\u6570<\/strong>(k\/\u221e)<\/li>\n<li><strong>\u96f6\u4e4b\u95f4\u7684\u96f6\u4e0d\u786e\u5b9a\u6027<\/strong>(0\/0)<\/li>\n<li><strong>\u65e0\u7a77\u5927 (\u65e0\u7a77\u5927\/\u65e0\u7a77\u5927) \u4e4b\u95f4\u7684\u65e0\u9650\u4e0d\u786e\u5b9a\u6027<\/strong><\/li>\n<li><strong>\u4e0d\u786e\u5b9a\u6027 1 \u63d0\u9ad8\u5230\u65e0\u7a77\u5927<\/strong>(1 <sup>\u221e<\/sup> )<\/li>\n<li><strong>\u96f6\u4e0d\u786e\u5b9a\u6027\u63d0\u5347\u81f3\u96f6<\/strong>(0 <sup>0<\/sup> )<\/li>\n<li> <strong style=\"font-family: -apple-system, BlinkMacSystemFont, &quot;Segoe UI&quot;, Roboto, Oxygen-Sans, Ubuntu, Cantarell, &quot;Helvetica Neue&quot;, sans-serif; font-size: 1rem;\">\u65e0\u7a77\u5927\u7684\u96f6\u4e0d\u786e\u5b9a\u6027<\/strong><span style=\"font-family: -apple-system, BlinkMacSystemFont, &quot;Segoe UI&quot;, Roboto, Oxygen-Sans, Ubuntu, Cantarell, &quot;Helvetica Neue&quot;, sans-serif; font-size: 1rem; font-weight: inherit;\">(0\u00b7\u221e)<\/span><\/li>\n<li><strong>\u96f6\u4e0d\u786e\u5b9a\u6027\u63d0\u5347\u81f3\u65e0\u7a77\u5927<\/strong>(0 <sup>\u221e<\/sup> )<\/li>\n<li><strong>\u65e0\u9650\u4e0d\u786e\u5b9a\u6027\u4e3a\u96f6<\/strong>(\u221e <sup>0<\/sup> )<\/li>\n<\/ul>\n<p>\u7136\u540e\u6211\u4eec\u5c06\u770b\u5230\u5982\u4f55\u89e3\u51b3\u6240\u6709\u7c7b\u578b\u7684\u4e0d\u786e\u5b9a\u6027\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><strong>\u4e0d\u5b9a\u5f62\u5f0f\u65e0\u7a77\u5927\u51cf\u65e0\u7a77\u5927<\/strong>\u4e0d\u7b49\u4e8e\u96f6\uff0c\u56e0\u4e3a\u6211\u4eec\u51cf\u53bb\u4e24\u4e2a\u975e\u5e38\u5927\u7684\u6570\uff0c\u4f46\u6211\u4eec\u4e0d\u77e5\u9053\u54ea\u4e2a\u66f4\u5927\u3002\u56e0\u6b64\uff0c\u65e0\u7a77\u5927\u4e4b\u5dee\u7684\u7ed3\u679c\u53d6\u51b3\u4e8e\u6bcf\u4e2a\u65e0\u7a77\u5927\u7684\u9636\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-03349653243a9ad62377c721fea0e797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\infty-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u89e3\u51b3\u8fd9\u79cd\u7c7b\u578b\u7684\u4e0d\u786e\u5b9a\u6027\u5e76\u4e0d\u5bb9\u6613\uff0c\u56e0\u4e3a\u6839\u636e\u51fd\u6570\u7684\u7c7b\u578b\uff0c\u5fc5\u987b\u5e94\u7528\u4e00\u79cd\u6216\u53e6\u4e00\u79cd\u8fc7\u7a0b\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u5efa\u8bae\u60a8\u5728\u4ee5\u4e0b\u94fe\u63a5\u4e2d\u67e5\u770b\u5b8c\u6574\u7684\u8bf4\u660e\uff1a<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span><strong>\u8bf7\u53c2\u9605\uff1a<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/cn\/\u4e0d\u786e\u5b9a\u6027-\u65e0\u9650\u51cf\u65e0\u9650-%e2%88%9e-%e2%88%9e\/\">\u5982\u4f55\u89e3\u51b3\u4e0d\u786e\u5b9a\u6027\u65e0\u7a77\u5927\u51cf\u53bb\u65e0\u7a77\u5927<\/a><\/span><\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-numero-entre-cero\"><\/span>\u96f6\u4e4b\u95f4\u7684\u4e0d\u786e\u5b9a\u6570<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u5f53\u6709\u7406\u51fd\u6570\u7684\u5206\u6bcd\u88ab\u53d6\u6d88\u65f6\uff0c\u5f97\u5230<strong>\u5e38\u6570\u9664\u4ee5\u96f6\u7684\u4e0d\u786e\u5b9a\u6027<\/strong>\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-8d550f72be2a531eb89d5cf200f54dc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{k}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"10\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u8fd9\u79cd\u4e0d\u5b9a\u5f62\u5f0f\u7684\u7ed3\u679c\u603b\u662f\u66f4\u5927\u3001\u66f4\u5c0f\u6216\u8005\u51fd\u6570\u7684\u6781\u9650\u4e0d\u5b58\u5728\u3002\u8ba9\u6211\u4eec\u4ee5\u6c42\u89e3\u6781\u9650\u4e3a\u4f8b\u6765\u770b\u770b\u5982\u4f55\u8ba1\u7b97\u8fd9\u79cd\u4e0d\u786e\u5b9a\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-f20a823489187682e3becf93cbd93c7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\cfrac{-4}{x^2}=\\cfrac{-4}{0^2}=\\cfrac{-4}{0}=\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"217\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u5df2\u7ecf\u5f97\u5230\u4e86\u4e00\u4e2a\u6570\u9664\u4ee5\u96f6\u7684\u4e0d\u786e\u5b9a\u6027\uff0c<u style=\"text-decoration-color:#FF9B28;\">\u56e0\u6b64\u6211\u4eec\u9700\u8981\u8ba1\u7b97\u51fd\u6570\u7684\u6a2a\u5411\u6781\u9650\uff1a<\/u><\/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-84821ecfa11641959a1463c3f2dd00e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0^-}\\cfrac{-4}{0^2}=\\cfrac{-4}{(-0,001)^2}=\\cfrac{-4}{+0}=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"289\" style=\"vertical-align: -17px;\"><\/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-904f4d1bb401493cdb76cbb9ce607f0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0^+}\\cfrac{-4}{0^2}=\\cfrac{-4}{0,001^2}=\\cfrac{-4}{+0}=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"261\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span><strong>\u8bf7\u53c2\u9605\uff1a<\/strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/cn\/\u6a2a\u5411\u9650\u5236\/\">\u4ec0\u4e48\u662f\u6a2a\u5411\u9650\u5236\uff1f<\/a><\/span><\/p>\n<p>\u51fd\u6570\u7684\u4e24\u4e2a\u6a2a\u5411\u6781\u9650\u7ed9\u51fa\u76f8\u540c\u7684\u7ed3\u679c\uff0c\u56e0\u6b64\u6839\u636e\u5b9a\u4e49\uff0c\u5f53 x \u8d8b\u4e8e 0 \u65f6\u51fd\u6570\u7684\u6781\u9650\u7ed9\u51fa\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-9c1b718f44360fe4322ba69ee40c9613_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)=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"401\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p>\u8bf7\u6ce8\u610f\uff0c\u5982\u679c\u6a2a\u5411\u6781\u9650\u7ed9\u51fa\u4e86\u4e0d\u540c\u7684\u503c\uff0c\u5219\u6b64\u65f6\u51fd\u6570\u7684\u6781\u9650\u5c06\u4e0d\u5b58\u5728\u3002 <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-cero-entre-cero\"><\/span>\u96f6\u4e4b\u95f4\u7684\u4e0d\u786e\u5b9a\u6027<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>\u4e0d\u786e\u5b9a\u6781\u9650\u96f6\u9664\u4ee5\u96f6<\/strong>\u662f\u5f88\u5e38\u89c1\u7684\uff0c\u5e76\u4e14\u662f\u5728\u5206\u5b50\u548c\u5206\u6bcd\u62b5\u6d88\u7684\u5206\u6570\u51fd\u6570\u4e2d\u83b7\u5f97\u7684\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-f88512bac0562399d5d8e65829073b54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"9\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u6839\u636e\u51fd\u6570\u7684\u4e0d\u540c\uff0c\u8fd9\u79cd\u7c7b\u578b\u7684\u4e0d\u786e\u5b9a\u6781\u9650\u7684\u89e3\u51b3\u65b9\u5f0f\u4e5f\u4e0d\u540c\u3002\u4f8b\u5982\uff0c\u5982\u679c\u51fd\u6570\u6709\u6839\uff0c\u5219\u5fc5\u987b\u6267\u884c\u4e0d\u540c\u7684\u6b65\u9aa4\u3002\u60a8\u53ef\u4ee5\u5728\u4ee5\u4e0b\u94fe\u63a5\u4e2d\u67e5\u770b\u6b64\u7c7b\u4e0d\u786e\u5b9a\u6027\u7684\u4e0d\u540c\u89e3\u51b3\u65b9\u6848\uff1a<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span><strong>\u53c2\u89c1\uff1a<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/cn\/\u96f6\u4e4b\u95f4-\u96f6-0-0-\u4e0d\u786e\u5b9a\/\">\u5982\u4f55\u89e3\u51b3\u96f6\u4e4b\u95f4\u7684\u96f6\u4e0d\u786e\u5b9a\u6027<\/a><\/span><\/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><strong>\u65e0\u7a77\u5927\u4e4b\u95f4\u7684\u65e0\u9650\u4e0d\u786e\u5b9a\u6027<\/strong>\u901a\u5e38\u51fa\u73b0\u5728\u5e26\u6709\u5206\u6570\u7684\u51fd\u6570\u7684\u65e0\u7a77\u6781\u9650\u4e2d\u3002\u5c3d\u7ba1\u4e0d\u786e\u5b9a\u6027\u662f\u4e24\u4e2a\u65e0\u7a77\u5927\u7684\u5546\uff0c\u4f46\u7ed3\u679c\u4e0d\u4e00\u5b9a\u662f\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-5322af410095265a81aa545e533ebd1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\infty}{\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"18\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u8fd9\u79cd\u4e0d\u786e\u5b9a\u5f62\u5f0f\u662f\u901a\u8fc7\u6bd4\u8f83\u6765\u89e3\u51b3\u7684\u3002\u5373\uff0c\u89c2\u5bdf\u5206\u5b50\u7684\u6b21\u6570\u548c\u5206\u6bcd\u7684\u6b21\u6570\uff0c\u6839\u636e\u54ea\u4e00\u4e2a\u8f83\u5927\uff0c\u6781\u9650\u7ed3\u679c\u662f\u5176\u4e2d\u4e4b\u4e00\u6216\u53e6\u4e00\u4e2a\u3002\u60a8\u53ef\u4ee5\u5728\u4ee5\u4e0b\u94fe\u63a5\u4e2d\u67e5\u770b\u6240\u6709\u6848\u4f8b\uff1a<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span><strong>\u8bf7\u53c2\u9605\uff1a<\/strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/cn\">\u5df2\u89e3\u51b3\u7684\u5173\u4e8e\u65e0\u7a77\u5927\u4e4b\u95f4\u7684\u65e0\u9650\u6781\u9650\u7684\u7ec3\u4e60<\/a><\/span><\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-1-elevado-a-infinito\"><\/span>\u4e0d\u786e\u5b9a\u6027 1 \u63d0\u9ad8\u5230\u65e0\u7a77\u5927<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u4ece\u6570\u5b66\u4e0a\u8bb2\uff0c\u4eba\u4eec\u53ef\u80fd\u4f1a\u8ba4\u4e3a<strong>1 \u5230\u65e0\u7a77\u5927<\/strong>\u7b49\u4e8e 1\uff0c\u56e0\u4e3a 1 \u7684\u4efb\u4f55\u5e42\u90fd\u7b49\u4e8e 1\u3002\u7136\u800c\uff0c\u8fd9\u9879\u662f\u4e0d\u786e\u5b9a\u6027\u7684\uff0c\u56e0\u6b64\uff0c\u4eba\u4eec\u4e0d\u80fd\u8f7b\u6613\u5730\u63a8\u5bfc\u51fa\u5b83\u7684\u7ed3\u679c\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-f68a3b617b5f1db37a9beea43e15264f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1^{\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u8fd9\u79cd\u7c7b\u578b\u7684\u4e0d\u786e\u5b9a\u6027\u901a\u8fc7\u5e94\u7528\u4ee5\u4e0b\u516c\u5f0f\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-e6bc13731241df168c3dc37d3e3b8e58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to+\\infty}f(x)^{g(x)}=\\lim_{x\\to+\\infty}e^{g(x)\\cdot [f(x)-1]}\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"271\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p>\u4f8b\u5982\uff0c\u4ee5\u4e0b\u6781\u9650\u662f\u4e0d\u786e\u5b9a\u7684\uff0c\u56e0\u4e3a\u5b83\u7ed9\u51fa\u4e86\u65e0\u7a77\u5927\u7684\u5e42\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-96004b3ff4fd888f2e7fbc30a14abf13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to+\\infty}\\left(1-\\frac{1}{x}\\right)^x=\\left(1-\\frac{1}{+\\infty}\\right)^{+\\infty}=(1-0)^{+\\infty}=1^{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"428\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u6211\u4eec\u5fc5\u987b\u4f7f\u7528\u6b64\u7c7b\u4e0d\u786e\u5b9a\u6027\u7684\u516c\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-e577daa5f0e768ca16bfe27a4d7a8a85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to+\\infty}e^{x\\cdot\\left[1-\\frac{1}{x}-1\\right]}=\\lim_{x\\to+\\infty}e^{x\\cdot\\left[-\\frac{1}{x}\\right]}=\\lim_{x\\to+\\infty}e^{-1}=\\frac{1}{e}\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"383\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\u6211\u4eec\u5df2\u7ecf\u89e3\u51b3\u4e86\u5c06\u4e0d\u786e\u5b9a\u6781\u9650\u63d0\u9ad8\u5230\u65e0\u7a77\u5927\u7684\u95ee\u9898\u3002 <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-cero-elevado-a-cero\"><\/span>\u96f6\u4e0d\u786e\u5b9a\u6027\u5f52\u96f6<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>\u96f6\u5e42\u7684\u96f6\u4e0d\u786e\u5b9a\u6027<\/strong>\u51fa\u73b0\u5728\u590d\u6742\u51fd\u6570\u7684\u6781\u9650\u5185\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-dfc1e32d3dc765b27701a8576e765fc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"0^0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u8981\u89e3\u51b3\u8fd9\u79cd\u7c7b\u578b\u7684\u4e0d\u786e\u5b9a\u9650\u5236\uff0c\u60a8\u5fc5\u987b\u4f7f\u7528\u4ee5\u4e0b\u9650\u5236\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-56b621d520e28cf2dbcd93bcd5d35eb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a}f(x)^{g(x)}=e^{^{\\displaystyle\\lim_{x\\to a}g(x)\\cdot \\ln\\bigl(f(x)\\bigr)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"266\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u4f8b\u5982\uff0c\u4ee5\u4e0b\u6781\u9650\u7ed9\u51fa\u4e86\u4e0d\u5b9a\u5f62\u5f0f 0 \u7684 0 \u6b21\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-d8680b80b19cae955169d2c0a8ad41f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(\\frac{1}{x}\\right)^{\\frac{1}{x}}=\\left(\\frac{1}{+\\infty}\\right)^{\\frac{1}{+\\infty}}=0^0\" title=\"Rendered by QuickLaTeX.com\" height=\"50\" width=\"251\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u4f46\u5982\u679c\u6211\u4eec\u5bf9\u6781\u9650\u53d6\u5bf9\u6570\uff0c\u6211\u4eec\u53ef\u4ee5\u627e\u5230\u5b83\u7684\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-04165b15f4b40bbe84ae5a4b214d4846_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to +\\infty}\\left(\\frac{1}{x}\\right)^{\\frac{1}{x}}=e^{^{\\displaystyle\\lim_{x\\to +\\infty}\\frac{1}{x}\\cdot \\ln\\left(\\frac{1}{x}\\right)}}=\\\\[5ex]\\displaystyle =e^{^{\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\ln\\left(\\frac{1}{x}\\right)}{x}}}=e^{^{\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\ln1-\\ln x}{x}}}=\\\\[5ex]=\\displaystyle e^{^{\\displaystyle\\lim_{x\\to +\\infty}\\frac{-\\ln x}{x}}}=e^{^{\\displaystyle\\frac{-\\infty}{+\\infty}}}=e^0=1\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"239\" width=\"307\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-cero-por-infinito\"><\/span>\u65e0\u7a77\u5927\u7684\u96f6\u4e0d\u786e\u5b9a\u6027<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>\u96f6\u548c\u65e0\u7a77\u5927\u7684\u4e58\u79ef\u7684\u4e0d\u786e\u5b9a\u6027<\/strong>\u5f88\u96be\u9047\u5230\uff0c\u4f46\u8fd9\u5e76\u4e0d\u610f\u5473\u7740\u5b83\u5f88\u5bb9\u6613\u786e\u5b9a\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-74b7fdba8e0feae988551f83341ec063_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"0\\cdot \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u6ca1\u6709\u5355\u4e00\u7684\u65b9\u6cd5\u53ef\u4ee5\u89e3\u51b3\u8fd9\u79cd\u7c7b\u578b\u7684\u4e0d\u786e\u5b9a\u6027\uff0c\u4f46\u5b83\u53d6\u51b3\u4e8e\u51fd\u6570\u7684\u7c7b\u578b\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5fc5\u987b\u5c06\u51fd\u6570\u8f6c\u6362\u4e3a\u65e0\u9650\u4e0d\u786e\u5b9a\u6027\u9664\u4ee5\u65e0\u7a77\u5927\u6216\u96f6\u4e0d\u786e\u5b9a\u6027\u9664\u4ee5\u96f6\uff0c\u5e76\u4ece\u90a3\u91cc\u5e94\u7528\u6211\u4eec\u4e0a\u9762\u770b\u5230\u7684\u6bcf\u4e2a\u4e0d\u786e\u5b9a\u6027\u7684\u6c42\u89e3\u65b9\u6cd5\u3002<\/p>\n<p>\u56e0\u6b64\uff0c\u5982\u679c\u4e00\u4e2a\u51fd\u6570\u7684\u6781\u9650\u4e3a 0\uff0c\u53e6\u4e00\u4e2a\u51fd\u6570\u7684\u6781\u9650\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-ac028c501a7835fdfa3bab5c769849b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a}f(x)=0\\qquad\\lim_{x\\to a}g(x)=\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"243\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8fdb\u884c\u4ee5\u4e0b\u66f4\u6539\u6765\u65e0\u9650\u5730\u8f6c\u6362\u6b64\u7c7b\u578b\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-df402461269ae26c30768fc0bf83f2ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a}f(x)\\cdot g(x)\\begin{cases}\\displaystyle\\lim_{x\\to a}\\frac{f(x)}{\\displaystyle\\frac{1}{g(x)}}=\\frac{0}{0}\\\\[10ex]\\displaystyle\\lim_{x\\to a}\\frac{g(x)}{\\displaystyle\\frac{1}{f(x)}}=\\frac{\\infty}{\\infty}\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"174\" width=\"248\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u8ba9\u6211\u4eec\u4ee5\u89e3\u51b3\u4e0d\u786e\u5b9a\u6781\u9650\u4e3a\u4f8b\u6765\u770b\u770b\u5982\u4f55\u505a\u5230\u8fd9\u4e00\u70b9\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-929e74562ababa44a253522d4474afad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}e^{-x}\\cdot x=e^{-\\infty}\\cdot (+\\infty)=0\\cdot \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"277\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u5bf9\u51fd\u6570\u8fdb\u884c\u8fd0\u7b97\u4ee5\u83b7\u5f97\u65e0\u7a77\u5927\u7684\u65e0\u9650\u4e0d\u786e\u5b9a\u6027\uff0c\u7136\u540e\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-3c29bbb439514449cd12fd8d66e327af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to +\\infty}e^{-x}\\cdot x=\\lim_{x\\to +\\infty}\\frac{x}{\\displaystyle\\frac{1}{e^{-x}}}=\\\\[6ex]=\\displaystyle \\lim_{x\\to +\\infty}\\frac{x}{e^x}=\\frac{+\\infty}{e^{+\\infty}}=\\frac{+\\infty}{+\\infty}=0\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"241\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-cero-elevado-a-infinito\"><\/span>\u96f6\u4e0d\u786e\u5b9a\u6027\u63d0\u9ad8\u5230\u65e0\u7a77\u5927<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>\u5c06\u4e0d\u786e\u5b9a\u6027\u96f6\u63d0\u5347\u5230\u65e0\u7a77\u5927<\/strong>\u6709\u70b9\u96be\u4ee5\u7406\u89e3\uff0c\u56e0\u4e3a\u6211\u4eec\u6b63\u5728\u5c06\u4e00\u4e2a\u975e\u5e38\u5c0f\u7684\u6570\u5b57\u63d0\u5347\u5230\u4e00\u4e2a\u975e\u5e38\u5927\u7684\u6570\u5b57\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-4894b918853d6dca8db65a026b1a349b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"0^{\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u5f53\u83b7\u5f97\u8fd9\u4e9b\u4e0d\u5b9a\u5f62\u5f0f\u65f6\uff0c\u5fc5\u987b\u4f7f\u7528\u4ee5\u4e0b\u516c\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-56b621d520e28cf2dbcd93bcd5d35eb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a}f(x)^{g(x)}=e^{^{\\displaystyle\\lim_{x\\to a}g(x)\\cdot \\ln\\bigl(f(x)\\bigr)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"266\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u8ba9\u6211\u4eec\u89e3\u51b3\u4e00\u4e2a\u4f8b\u5b50\uff0c\u4ee5\u66f4\u597d\u5730\u7406\u89e3\u5982\u4f55\u8ba1\u7b97\u8fd9\u79cd\u7c7b\u578b\u7684\u4e0d\u786e\u5b9a\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-6ca354428ea8889a956a9b77b04a088f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 0^+}x^{\\frac{1}{x}}=e^{^{\\displaystyle\\lim_{x\\to 0^+}\\frac{1}{x}\\cdot \\ln(x)}}=\\\\[3.5ex]\\displaystyle =e^{^{\\displaystyle\\frac{1}{0^+}\\cdot \\ln(0^+)}}=e^{+\\infty\\cdot (-\\infty)}\\\\[3ex]\\displaystyle =e^{-\\infty}=\\frac{1}{e^{+\\infty}}=0\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"190\" width=\"221\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-infinito-elevado-a-cero\"><\/span>\u65e0\u9650\u4e0d\u786e\u5b9a\u6027\u5f52\u96f6<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u901a\u5e38\uff0c\u4efb\u4f55\u5e42\u4e3a\u96f6\u65f6\u90fd\u4f1a\u5f97\u5230 1\uff0c\u4f46\u662f\uff0c<strong>\u65e0\u7a77\u5927\u7684\u4e0d\u786e\u5b9a\u6027\u4e3a\u96f6\u65f6<\/strong>\u4e0d\u4e00\u5b9a\u662f\u8fd9\u6837\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-4a2ef9259a35e0e2f7a176bcdb934ad9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\infty^0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"25\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u6b63\u5982\u5728\u4e0d\u786e\u5b9a\u6027\u4e2d\uff0c\u96f6\u5347\u5230\u96f6\u548c\u96f6\u5347\u5230\u65e0\u7a77\u5927\uff0c\u8981\u89e3\u51b3\u8fd9\u79cd\u7c7b\u578b\u7684\u4e0d\u786e\u5b9a\u6781\u9650\uff0c\u6709\u5fc5\u8981\u5e94\u7528\u5bf9\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-56b621d520e28cf2dbcd93bcd5d35eb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a}f(x)^{g(x)}=e^{^{\\displaystyle\\lim_{x\\to a}g(x)\\cdot \\ln\\bigl(f(x)\\bigr)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"266\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u8ba9\u6211\u4eec\u901a\u8fc7\u9010\u6b65\u8ba1\u7b97\u793a\u4f8b\u6765\u770b\u770b\u5982\u4f55\u89e3\u51b3\u6b64\u7c7b\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-a45090015a206189aca3884f8b2cab30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to +\\infty}x^{\\frac{1}{x}}=e^{^{\\displaystyle\\lim_{x\\to +\\infty}\\frac{1}{x}\\cdot \\ln(x)}}=\\\\[3ex]\\displaystyle =e^{^{\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\ln(x)}{x}}}=e^{^{\\displaystyle\\frac{\\ln(+\\infty)}{+\\infty}}}=\\\\[3ex]\\displaystyle =e^{^{\\displaystyle\\frac{+\\infty}{+\\infty}}}=e^0=1\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"199\" width=\"236\" style=\"vertical-align: 0px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5728\u8fd9\u7bc7\u6587\u7ae0\u4e2d\uff0c\u6211\u4eec\u89e3\u91ca\u4ec0\u4e48\u662f\u4e0d\u786e\u5b9a\u6027\u3002\u60a8\u5c06\u53d1\u73b0\u6240\u6709\u7c7b\u578b\u7684\u4e0d\u786e\u5b9a\u6027\u662f\u4ec0\u4e48\u4ee5\u53ca\u5982\u4f55\u89e3\u51b3\u5b83\u4eec\u3002\u6b64\u5916\uff0c\u60a8\u5c06\u80fd\u591f\u770b\u5230\u9010\u6b65 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/cn\/%e4%b8%8d%e7%a1%ae%e5%ae%9a%e7%b1%bb%e5%9e%8b%e4%b8%8d%e7%a1%ae%e5%ae%9a%e6%9e%81%e9%99%90\/\"> <span class=\"screen-reader-text\">\u4e0d\u786e\u5b9a\u7684\u7c7b\u578b\uff08\u4e0d\u786e\u5b9a\u6781\u9650\uff09<\/span> \u67e5\u770b\u5168\u6587 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