{"id":17,"date":"2023-09-17T11:09:22","date_gmt":"2023-09-17T11:09:22","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e6%80%a7%e8%b4%a8%e6%9e%81%e9%99%90%e5%ae%9a%e5%be%8b\/"},"modified":"2023-09-17T11:09:22","modified_gmt":"2023-09-17T11:09:22","slug":"%e6%80%a7%e8%b4%a8%e6%9e%81%e9%99%90%e5%ae%9a%e5%be%8b","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e6%80%a7%e8%b4%a8%e6%9e%81%e9%99%90%e5%ae%9a%e5%be%8b\/","title":{"rendered":"\u6781\u9650\u6027\u8d28\uff08\u6216\u5b9a\u5f8b\uff09"},"content":{"rendered":"<p>\u5728\u8fd9\u91cc\u60a8\u5c06\u627e\u5230\u51fd\u6570\u6781\u9650\u7684\u6240\u6709\u5c5e\u6027\uff08\u6216\u5b9a\u5f8b\uff09\u3002\u8fd9\u4e9b\u5c5e\u6027\u7528\u4e8e\u7b80\u5316\u6781\u9650\u8ba1\u7b97\uff0c\u7279\u522b\u662f\u5728\u5904\u7406\u51fd\u6570\u8fd0\u7b97\u7684\u6781\u9650\u65f6\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcuales-son-las-propiedades-o-leyes-de-los-limites-de-funciones\"><\/span>\u51fd\u6570\u6781\u9650\u7684\u6027\u8d28\uff08\u6216\u5b9a\u5f8b\uff09\u662f\u4ec0\u4e48\uff1f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u89e3\u91ca\u51fd\u6570\u6781\u9650\u7684\u6240\u6709\u6027\u8d28\uff0c\u4e5f\u79f0\u4e3a\u51fd\u6570\u6781\u9650\u5b9a\u5f8b\u3002\u6b64\u5916\uff0c\u60a8\u5c06\u80fd\u591f\u770b\u5230\u6bcf\u4e2a\u6781\u9650\u5c5e\u6027\u7684\u5df2\u89e3\u51b3\u7ec3\u4e60\uff0c\u4ee5\u4fbf\u60a8\u53ef\u4ee5\u5b8c\u5168\u7406\u89e3\u8be5\u6982\u5ff5\u3002<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-suma\"><\/span>\u603b\u548c\u6781\u9650\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h3>\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>\u4e24\u4e2a\u51fd\u6570\u5728\u67d0\u4e00\u70b9\u7684\u6781\u9650\u4e4b<\/strong>\u548c\u7b49\u4e8e\u6bcf\u4e2a\u51fd\u6570\u5728\u8be5\u70b9\u7684\u6781\u9650\u4e4b\u548c\u3002<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e1975fcd0e98e2022e29be694fcdb925_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a} \\Bigl[ f(x)+g(x)\\Bigr]=\\lim_{x\\to a}f(x)+\\lim_{x\\to a}g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"310\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u4f8b\u5982\uff0c\u5047\u8bbe\u6709\u4e24\u4e2a\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-1730973da231e978e4c565c939633b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2\\qquad g(x)=2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"217\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> x \u7b49\u4e8e 1 \u65f6\u6bcf\u4e2a\u51fd\u6570\u7684\u6781\u9650\u4e3a\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-540347d33b8f0dc2d46cbc8d6f42df12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 1}x^2=1^2=1\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"121\" 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-9e8d60234a5aaad645bcea92331ffec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 1}(2x+1)=2\\cdot1+1=3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"209\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u4e24\u4e2a\u51fd\u6570\u5728\u540c\u4e00\u70b9\u76f8\u52a0\u7684\u6781\u9650\u4e3a 4 (1+3=4)\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-dee2fbc7ec3a4f0a145e79c1f8bf9ae2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 1} \\Bigl[ f(x)+g(x)\\Bigr]=\\\\[3ex]\\displaystyle =\\lim_{x\\to 1}f(x)+\\lim_{x\\to 1}g(x)=\\\\[3ex]=1+3=4\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"112\" width=\"189\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u8be5\u6027\u8d28\u53ef\u4ee5\u901a\u8fc7\u9010\u6b65\u8ba1\u7b97\u6781\u9650\u6765\u8bc1\u660e\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-14c858a555a3d57a24cc5c81adeda2fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 1} \\Bigl[ f(x)+g(x)\\Bigr]=\\\\[3ex]\\displaystyle =\\lim_{x\\to 1}\\Bigl[x^2+2x+1\\Bigr]=\\\\[3ex]=1^2+2\\cdot 1+1=4\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"117\" width=\"171\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-resta\"><\/span>\u51cf\u6cd5\u6781\u9650\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h3>\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>\u4e24\u4e2a\u51fd\u6570\u5728\u67d0\u4e00\u70b9\u76f8\u51cf\uff08\u6216\u5dee\uff09\u7684\u6781\u9650<\/strong>\u76f8\u5f53\u4e8e\u5206\u522b\u51cf\u53bb\u6bcf\u4e2a\u51fd\u6570\u5728\u8be5\u70b9\u7684\u6781\u9650\u3002<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-444c20b463064780542e57269cfd770e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a} \\Bigl[ f(x)-g(x)\\Bigr]=\\lim_{x\\to a}f(x)-\\lim_{x\\to a}g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"310\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u4f7f\u7528\u4e0a\u4e00\u4e2a\u793a\u4f8b\u4e2d\u7684\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-1730973da231e978e4c565c939633b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2\\qquad g(x)=2x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"217\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6bcf\u4e2a\u51fd\u6570\u5728 x=3 \u5904\u7684\u6781\u9650\u4e3a\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-f37267a3dc92ccef5a66ccfb68211919_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 3}x^2=3^2=9\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"122\" 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-64b82eccca7888756f96190e6374281a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 3}(2x+1)=2\\cdot3+1=7\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"209\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u90a3\u4e48\uff0c\u4e24\u4e2a\u51fd\u6570\u5728x=3\u5904\u76f8\u51cf\u7684\u6781\u9650\u5c31\u662f\u4e0a\u4e00\u6b65\u5f97\u5230\u7684\u503c\u7684\u5dee\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-85e829be6e72a7ab7bd91d442c89558f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 3} \\Bigl[ f(x)-g(x)\\Bigr]=\\\\[3ex]\\displaystyle =\\lim_{x\\to 3}f(x)-\\lim_{x\\to 3}g(x)=\\\\[3ex]=9-7=2\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"110\" width=\"189\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u51fd\u6570\u7684\u51cf\u6cd5\u7136\u540e\u6c42\u89e3\u6781\u9650\u6765\u8bc1\u660e\u6781\u9650\u7684\u6027\u8d28\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-411bbd6ad4e5d44322ec93ba06fd3088_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 3} \\Bigl[ f(x)-g(x)\\Bigr]=\\\\[3ex]\\displaystyle =\\lim_{x\\to 3}\\Bigl[x^2-(2x+1)\\Bigr]=\\\\[3ex]\\displaystyle =\\lim_{x\\to 3}\\Bigl[x^2-2x-1\\Bigr]\\\\[3ex]=3^2-2\\cdot 3-1=2\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"165\" width=\"185\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-de-limite-de-un-producto\"><\/span>\u9650\u5236\u4ea7\u54c1\u7684\u5c5e\u6027<span class=\"ez-toc-section-end\"><\/span><\/h3>\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>\u4e24\u4e2a\u51fd\u6570\u5728\u4e00\u70b9\u7684\u4e58\u79ef\u7684\u6781\u9650<\/strong>\u662f\u6bcf\u4e2a\u51fd\u6570\u5728\u8be5\u70b9\u7684\u6781\u9650\u7684\u4e58\u79ef\u3002<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d513f66b066a53d8dbd18868f1edcaa3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a} \\Bigl[ f(x)\\cdot g(x)\\Bigr]=\\lim_{x\\to a}f(x)\\cdot \\lim_{x\\to a}g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"292\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u4f8b\u5982\uff0c\u5982\u679c\u6211\u4eec\u6709\u4ee5\u4e0b\u4e24\u4e2a\u4e0d\u540c\u7684\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-9c48ccc2acf27d4fb1519d80b4c0cbe1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3\\qquad g(x)=x^2-5\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"216\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> x=2 \u5904\u6bcf\u4e2a\u51fd\u6570\u7684\u6781\u9650\u4e3a\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-eb87f9cfecaec2126043e52053704c82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 2}x^3=2^3=8\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"122\" 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-c62f8966fd75151d6bdc8955458d1557_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 2}(x^2-5)=2^2-5=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"207\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u8981\u786e\u5b9a\u4e24\u4e2a\u51fd\u6570\u7684\u4e58\u79ef\u7684\u6781\u9650\uff0c\u4e0d\u5fc5\u5c06\u5b83\u4eec\u76f8\u4e58\uff0c\u800c\u662f\u5c06\u6bcf\u4e2a\u6781\u9650\u5f97\u5230\u7684\u7ed3\u679c\u76f8\u4e58\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-0df37e873bf968f39617b8dce74edb9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 2} \\Bigl[ f(x)\\cdot g(x)\\Bigr]=\\\\[3ex]\\displaystyle =\\lim_{x\\to 2}f(x)\\cdot \\lim_{x\\to 2}g(x)=\\\\[3ex]=8\\cdot (-1)=-8\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"115\" width=\"180\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u8fd9\u8282\u7701\u4e86\u6211\u4eec\u7684\u65f6\u95f4\u548c\u8ba1\u7b97\u91cf\uff0c\u56e0\u4e3a\u4e24\u4e2a\u51fd\u6570\u76f8\u4e58\u53ef\u80fd\u5f88\u56f0\u96be\u3002 <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-un-cociente\"><\/span>\u5546\u6781\u9650\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h3>\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>\u4e24\u4e2a\u51fd\u6570\u7684\u5546\uff08\u6216\u9664\u6cd5\uff09\u7684\u6781\u9650<\/strong>\u7b49\u4e8e\u51fd\u6570\u6781\u9650\u7684\u5546\u3002<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dbb038d24d534af7fc12466761b1206c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a} \\left[\\frac{f(x)}{g(x)}\\right]=\\frac{\\displaystyle\\lim_{x\\to a}f(x)}{\\displaystyle\\lim_{x\\to a}g(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"181\" style=\"vertical-align: -24px;\"><\/p>\n<\/p>\n<p>\u53ea\u8981\u5206\u6bcd\u51fd\u6570\u7684\u6781\u9650\u4e0d\u4e3a\u96f6\uff0c\u5c31\u6ee1\u8db3\u6b64\u6761\u4ef6\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-615afb4e193ec9b817d1672cb66f67b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a}g(x)\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"97\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u5c06\u89e3\u51b3\u8fd9\u4e2a\u6781\u9650\u5c5e\u6027\uff08\u6216\u5b9a\u5f8b\uff09\u7684\u4f8b\u5b50\u3002\u8003\u8651\u51fd\u6570 f(x) \u548c g(x)\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-193a456033487c8b6501aa861a1b6351_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=5x-1\\qquad g(x)=3^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"217\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u9996\u5148\u8ba1\u7b97\u6bcf\u4e2a\u51fd\u6570\u5728 x=0 \u5904\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-a25aaef5c60bc849e2cfe326093a82fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 0}(5x-1)=5\\cdot 0-1=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"222\" 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-40be0b69270ff2804525ec6aa35a934c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 0}3^x=3^0=1\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"121\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u627e\u5230 x=0 \u5904\u4e24\u4e2a\u51fd\u6570\u76f8\u9664\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-e8558b623352036e3af2e68e72861b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 0} \\left[\\frac{f(x)}{g(x)}\\right]=\\frac{\\displaystyle\\lim_{x\\to 0}f(x)}{\\displaystyle\\lim_{x\\to 0}g(x)}=\\displaystyle\\frac{-1}{1}=-1\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"278\" style=\"vertical-align: -24px;\"><\/p>\n<\/p>\n<p>\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5e94\u7528\u8fd9\u4e2a\u6027\u8d28\u6765\u6c42\u89e3\u6781\u9650\uff0c\u56e0\u4e3a g(x) \u7684\u6781\u9650\u662f\u975e\u96f6\u7684\u3002 <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-constante\"><\/span>\u5e38\u6570\u6781\u9650\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h3>\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>\u5e38\u6570\u51fd\u6570\u7684\u6781\u9650<\/strong>\u603b\u662f\u4ea7\u751f\u5e38\u6570\u672c\u8eab\uff0c\u65e0\u8bba\u6781\u9650\u662f\u5728\u4ec0\u4e48\u70b9\u8ba1\u7b97\u7684\u3002<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-083ac97d4077fd32108f371887a6daef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a} k=k\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"74\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u8fd9\u4e2a\u5c5e\u6027\u5f88\u5bb9\u6613\u68c0\u67e5\uff0c\u4f8b\u5982\uff0c\u5982\u679c\u6211\u4eec\u6709\u4ee5\u4e0b\u5e38\u91cf\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-e9e41bd168999d7634188ca08496465a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=5\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u4ece\u903b\u8f91\u4e0a\u8bb2\uff0c\u5e38\u6570\u51fd\u6570\u5728\u4efb\u610f\u70b9\u7684\u6781\u9650\u662f 5\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-cc495627062fe120c4fba7c649b94ea8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 0}5=5\\qquad\\qquad\\lim_{x\\to 3}5=5\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"218\" 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-bf06e3dadceb69c684be4e04205744f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to -2}5=5\\qquad\\qquad\\lim_{x\\to 7}5=5\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"229\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-un-multiplo-constante\"><\/span>\u5e38\u6570\u500d\u6570\u6781\u9650\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u6839\u636e\u4e58\u79ef\u6781\u9650\u548c\u5e38\u6570\u6781\u9650\u7684\u6027\u8d28\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u4ee5\u4e0b\u6027\u8d28\uff1a <\/p>\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>\u51fd\u6570\u7684\u6781\u9650\u4e58\u4ee5\u5e38\u6570<\/strong>\u7b49\u4e8e\u8be5\u5e38\u6570\u4e0e\u51fd\u6570\u7684\u6781\u9650\u7684\u4e58\u79ef\u3002<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9e11e20d92c504cfb48a1a5c3747066f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a}\\Bigl[ k\\cdot f(x)\\Bigr]=k\\cdot\\lim_{x\\to a}f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"214\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u8bf7\u6ce8\u610f\u6211\u4eec\u5982\u4f55\u4f7f\u7528\u6b64\u5c5e\u6027\u7b80\u5316\u4ee5\u4e0b\u9650\u5236\u7684\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-422e706a1b7643d78c0d667e17ede188_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle \\lim_{x\\to 4} (2x^2-12x+10)=\\\\[3ex]\\displaystyle =\\lim_{x\\to 4}\\Bigl[2\\cdot(x^2-6x+5)\\Bigr]=\\\\[3ex]=\\displaystyle 2\\cdot\\lim_{x\\to 4}(x^2-6x+5)=\\\\[3ex]=2\\cdot (4^2-6\\cdot4+5)=\\\\[3ex]=2\\cdot (-3)=-6\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"206\" width=\"206\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-potencia\"><\/span>\u5e42\u6781\u9650\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h3>\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>\u4efb\u4f55\u51fd\u6570\u63d0\u5347\u5230\u6307\u6570\u7684\u6781\u9650<\/strong>\u76f8\u5f53\u4e8e\u8ba1\u7b97\u51fd\u6570\u7684\u6781\u9650\uff0c\u7136\u540e\u5c06\u6781\u9650\u7684\u7ed3\u679c\u63d0\u5347\u5230\u8be5\u6307\u6570\u3002<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02b8185a9eca6e85f4d3e69b41b09903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a}\\Bigl[f(x)^k\\Bigr]=\\left[\\lim_{x\\to a}f(x)\\right]^k\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"202\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u4f8b\u5982\uff0c\u7ebf\u6027\u51fd\u6570\u7684\u6781\u9650\u4e3a\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-73523d73bd77b077f64d5bf05bd0444c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 6}x=6\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"74\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u90a3\u4e48\uff0c\u4e8c\u6b21\u51fd\u6570\u7684\u6781\u9650\u53ef\u4ee5\u901a\u8fc7\u627e\u5230\u7ebf\u6027\u51fd\u6570\u7684\u6781\u9650\u7136\u540e\u5bf9\u7ed3\u679c\u6c42\u5e73\u65b9\u6765\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-051ef387f34fdcc7468d761e2d618f69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 6}\\Bigl[x^2\\Bigr]=\\left[\\lim_{x\\to 6}x\\right]^2=\\bigl[6\\bigr]^2=36\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"249\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-funcion-exponencial\"><\/span>\u6307\u6570\u51fd\u6570\u6781\u9650\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h3>\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>\u6307\u6570\u51fd\u6570\u7684\u6781\u9650<\/strong>\u7b49\u4e8e\u51fd\u6570\u4ee3\u6570\u8868\u8fbe\u5f0f\u6781\u9650\u7684\u51fd\u6570\u5e38\u6570\u3002<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a2fc0c424758574c4b22a93c3e5dfdcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a}\\Bigl[k^{g(x)}\\Bigr]=k^{^{\\displaystyle\\lim_{x\\to a}g(x)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"179\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u7136\u540e\uff0c\u6211\u4eec\u5c06\u901a\u8fc7\u4e24\u79cd\u53ef\u80fd\u7684\u65b9\u5f0f\u8ba1\u7b97\u6307\u6570\u51fd\u6570\u7684\u6781\u9650\u6765\u9a8c\u8bc1\u6b64\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-6eb8d552dbdff46188019ca7d717e5a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1}5^{2x}=5^{2\\cdot 1}=25\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"147\" 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-1088a10d675c2a6dc332c23a362fce2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1}5^{2x}=5^{^{\\displaystyle\\lim_{x\\to 1}2x}}=5^{2\\cdot 1}=25\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"232\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-potencia-de-funciones\"><\/span>\u51fd\u6570\u5e42\u6781\u9650\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h3>\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>\u4e00\u4e2a\u51fd\u6570\u5347\u5230\u53e6\u4e00\u4e2a\u51fd\u6570\u7684\u6781\u9650<\/strong>\u662f\u7b2c\u4e00\u4e2a\u51fd\u6570\u7684\u6781\u9650\u5347\u5230\u7b2c\u4e8c\u4e2a\u51fd\u6570\u7684\u6781\u9650\u3002<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-11395cf067e1c7ff75ae68090bec8d16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to a}\\Bigl[f(x)^{g(x)}\\Bigr]=\\left[\\lim_{x\\to a}f(x)\\right]^{\\displaystyle\\lim_{x\\to a}g(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"277\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4f8b\u5982\uff0c\u6211\u4eec\u5c06\u5e94\u7528\u8be5\u6cd5\u5219\u786e\u5b9a\u4ee5\u4e0b\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-88cf4a1a6c04abab530e67f4f0ca950d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 2}\\Bigl[(x^2-4x)^{4x-5}\\Bigr]=\\\\[3ex]\\displaystyle =\\left[\\lim_{x\\to 2}(x^2-4x)\\right]^{\\displaystyle\\lim_{x\\to 2}(4x-5)}=\\\\[3ex]=\\displaystyle (2^2-4\\cdot 2)^{4\\cdot 2-5}=\\\\[3ex]=(-4)^3=-64\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"173\" width=\"247\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-funcion-irracional\"><\/span>\u65e0\u7406\u51fd\u6570\u6781\u9650\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h3>\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>\u6839\uff08\u6216\u6839\u5f0f\uff09\u7684\u6781\u9650<\/strong>\u7b49\u4e8e\u6781\u9650\u7684\u6839\u3002<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2cdf7e1a72f2441f72240f068edc413e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a}\\sqrt[n]{f(x)}=\\sqrt[n]{\\lim_{x\\to a}f(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"197\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p>\u8981\u4f7f\u7528\u6b64\u5c5e\u6027\uff0c\u5fc5\u987b\u8bb0\u4f4f\uff0c\u5982\u679c\u6839\u7d22\u5f15\u4e3a\u5076\u6570\uff0c\u5219\u51fd\u6570\u7684\u6781\u9650\u5fc5\u987b\u5927\u4e8e\u6216\u7b49\u4e8e 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-31077c9693c2318fba7cb2f0f99d5efd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{si } n \\text{ es par} \\ \\longrightarrow \\ \\displaystyle\\lim_{x\\to a}f(x)\\ge 0\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"230\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u8bf7\u6ce8\u610f\u5982\u4f55\u5e94\u7528\u6b64\u516c\u5f0f\u8ba1\u7b97\u4ee5\u4e0b\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-ad822bfcdd6fbb0335e1d99db71776c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 4}\\sqrt[3]{\\frac{x^2}{2}}=\\sqrt[3]{\\lim_{x\\to 4}\\frac{x^2}{2}}=\\sqrt[3]{\\frac{4^2}{2}}=\\sqrt[3]{8}=2\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"310\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedad-del-limite-de-una-funcion-logaritmica\"><\/span>\u5bf9\u6570\u51fd\u6570\u6781\u9650\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h3>\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>\u5bf9\u6570\u7684\u6781\u9650<\/strong>\u7b49\u4e8e\u6781\u9650\u7684\u540c\u5e95\u5bf9\u6570\u3002<\/p>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6e34ee6ca0162d42714f0c8f3e245b14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a}\\Bigl[\\log_k f(x)\\Bigr]=\\log_k \\left[\\lim_{x\\to a}f(x)\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"252\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u770b\u770b\u6211\u4eec\u5e94\u7528\u6b64\u5c5e\u6027\u7684\u4ee5\u4e0b\u9650\u5236\u7684\u5206\u8fa8\u7387\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" 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