{"id":364,"date":"2023-07-04T05:10:26","date_gmt":"2023-07-04T05:10:26","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e6%a8%aa%e5%90%91%e9%99%90%e5%88%b6\/"},"modified":"2023-07-04T05:10:26","modified_gmt":"2023-07-04T05:10:26","slug":"%e6%a8%aa%e5%90%91%e9%99%90%e5%88%b6","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e6%a8%aa%e5%90%91%e9%99%90%e5%88%b6\/","title":{"rendered":"\u6a2a\u5411\u9650\u5236"},"content":{"rendered":"<p>\u5728\u672c\u6587\u4e2d\uff0c\u6211\u4eec\u5c06\u89e3\u91ca\u4ec0\u4e48\u662f\u51fd\u6570\u7684\u6a2a\u5411\u6781\u9650\uff08\u5e76\u9644\u6709\u793a\u4f8b\uff09\u3002\u6211\u4eec\u8fd8\u6559\u60a8\u5982\u4f55\u4ee5\u56fe\u5f62\u548c\u6570\u5b57\u65b9\u5f0f\u8ba1\u7b97\u51fd\u6570\u7684\u5de6\u4fa7\u548c\u53f3\u4fa7\u6781\u9650\u3002\u6b64\u5916\uff0c\u60a8\u5c06\u80fd\u591f\u901a\u8fc7\u9010\u6b65\u89e3\u51b3\u6a2a\u5411\u9650\u5236\u7684\u7ec3\u4e60\u8fdb\u884c\u8bad\u7ec3\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-los-limites-laterales\"><\/span>\u6a2a\u5411\u9650\u5236\u662f\u591a\u5c11\uff1f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>\u51fd\u6570\u5728\u67d0\u4e00\u70b9\u7684\u6a2a\u5411\u6781\u9650<\/strong>\u7814\u7a76\u51fd\u6570\u5728\u8be5\u70b9\u5468\u56f4\u7684\u884c\u4e3a\u3002\u6709\u5de6\u4fa7\u6781\u9650\u548c\u53f3\u4fa7\u6781\u9650\uff0c\u5206\u522b\u5206\u6790\u51fd\u6570\u5728\u8003\u8651\u70b9\u5de6\u4fa7\u548c\u53f3\u4fa7\u7684\u503c\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"limites-laterales-por-la-izquierda-y-por-la-derecha\"><\/span>\u5de6\u53f3\u4fa7\u5411\u9650\u5236<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6b63\u5982\u6211\u4eec\u5728\u6a2a\u5411\u8fb9\u754c\u7684\u5b9a\u4e49\u4e2d\u770b\u5230\u7684\uff0c\u6709\u4e24\u79cd\u7c7b\u578b\uff1a\u5de6\u4fa7\u8fb9\u754c\u548c\u53f3\u4fa7\u8fb9\u754c\u3002<\/p>\n<p>\u51fd\u6570\u7684\u5de6\u6781\u9650\u5728\u5206\u6790\u6781\u9650\u7684\u70b9\u5904\u7528\u51cf\u53f7\u8868\u793a\uff0c\u800c\u53f3\u6781\u9650\u5219\u7528\u52a0\u53f7\u8868\u793a\u3002 <\/p>\n<div class=\"wp-block-columns are-vertically-aligned-center is-layout-flex wp-container-129\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><strong><u style=\"text-decoration-color:#FF9B28;\">\u5de6\u4fa7\u6a2a\u5411\u6781\u9650<\/u><\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3224554094b2418a485786bfbe4db5f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a^{\\color{orange}\\bm{-}\\color{black}}}f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"141\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><strong><u style=\"text-decoration-color:#FF9B28;\">\u53f3\u4fa7\u6a2a\u5411\u6781\u9650<\/u><\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9e2e49ad15a0b5fd2db93797689ea1b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a^{\\color{orange}\\bm{+}\\color{black}}}f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"141\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p>\u770b\u4e0b\u9762\u7684\u4f8b\u5b50\u53ef\u4ee5\u66f4\u597d\u5730\u7406\u89e3\u6a2a\u5411\u9650\u5236\u7684\u542b\u4e49\uff1a <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/limites-laterales.webp\" alt=\"\u6a2a\u5411\u9650\u5236\" class=\"wp-image-823\" width=\"299\" height=\"314\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p>\u6b63\u5982\u60a8\u5728\u6b64\u5206\u6bb5\u51fd\u6570\u7684\u56fe\u5f62\u8868\u793a\u4e2d\u6240\u770b\u5230\u7684\uff0c\u6a2a\u5411\u9650\u5236\u53d6\u51b3\u4e8e\u8ba1\u7b97\u5b83\u4eec\u7684\u4e00\u4fa7\u3002<\/p>\n<p>\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5f53 x \u4ece\u5de6\u8fb9\u63a5\u8fd1 2 \u65f6\uff0c\u51fd\u6570\u4f1a\u63a5\u8fd1 3\uff0c\u56e0\u4e3a\u5f53<em>x<\/em>\u4ece\u5de6\u8fb9\u63a5\u8fd1 x=2 \u65f6\uff0c\u51fd\u6570\u4f1a\u53d6\u66f4\u63a5\u8fd1 3 \u7684\u503c\u3002<\/p>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u51fd\u6570\u5728 x=2 \u5904\u7684\u76f4\u7ebf\u7684\u6a2a\u5411\u6781\u9650\u503c\u4e3a 6\u3002\u56e0\u4e3a\u5982\u679c\u6211\u4eec\u901a\u8fc7\u76f4\u7ebf\u63a5\u8fd1\u70b9 x=2\uff0c\u51fd\u6570\u7684\u503c\u4f1a\u8d8a\u6765\u8d8a\u63a5\u8fd1 f(x)= 6.<\/p>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u60a8\u5e94\u8be5\u77e5\u9053\u6a2a\u5411\u9650\u5236\u4e0e\u666e\u901a\u9650\u5236\u5177\u6709\u76f8\u540c\u7684\u5c5e\u6027\u3002\u5728\u4ee5\u4e0b\u94fe\u63a5\u4e2d\uff0c\u60a8\u53ef\u4ee5\u770b\u5230\u8fb9\u754c\u5c5e\u6027\u662f\u4ec0\u4e48\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\/\u6027\u8d28\u6781\u9650\u5b9a\u5f8b\/\">\u8fb9\u754c\u5c5e\u6027<\/a><\/span><\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"limites-laterales-iguales\"><\/span>\u7b49\u6a2a\u5411\u9650\u5236<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6211\u4eec\u521a\u521a\u770b\u5230\u4e86\u4e00\u4e2a\u51fd\u6570\u7684\u8fb9\u9650\u4e0d\u540c\u7684\u4f8b\u5b50\uff0c\u4f46\u662f&#8230;&#8230;\u5982\u679c\u8fb9\u9650\u76f8\u540c\u4f1a\u53d1\u751f\u4ec0\u4e48\uff1f<\/p>\n<p><strong>\u5982\u679c\u51fd\u6570\u5728\u4e00\u70b9\u5904\u7684\u4e24\u4e2a\u6a2a\u5411\u6781\u9650\u90fd\u5b58\u5728\u5e76\u4e14\u76f8\u7b49<\/strong>\uff0c\u5219\u51fd\u6570\u5728\u8be5\u70b9\u5b58\u5728\u6781\u9650\uff0c\u6781\u9650\u7684\u7ed3\u679c\u5c31\u662f\u6a2a\u5411\u6781\u9650\u7684\u503c\u3002<\/p>\n<p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u8981\u4f7f\u51fd\u6570\u5728\u4e00\u70b9\u5904\u5b58\u5728\u6781\u9650\uff0c\u5fc5\u987b\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\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-1d596fffab33786b9af4466210642acf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to a^-}f(x)=\\lim_{x\\to a^+}f(x)=L \\ \\iff \\ \\lim_{x\\to a}f(x)=L\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"378\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u5982\u679c\u51fd\u6570\u5728\u4e00\u70b9\u5904\u7684\u6a2a\u5411\u6781\u9650\u4e0d\u540c\uff0c\u5219\u8be5\u51fd\u6570\u5728\u8be5\u70b9\u7684\u6781\u9650\u4e0d\u5b58\u5728\u3002<\/p>\n<p>\u53e6\u5916\uff0c\u51fd\u6570\u5728\u70b9\u5904\u7684\u6781\u9650\u7684\u5b58\u5728\u662f<span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/cn\/\u8fde\u7eed\u51fd\u6570-\u51fd\u6570\u7684\u8fde\u7eed\u6027\/\">\u51fd\u6570\u5728\u70b9\u5904\u8fde\u7eed<\/a><\/span>\u7684\u5fc5\u8981\u6761\u4ef6\u3002<\/p>\n<p>\u8ba9\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u4f8b\u5b50\u6765\u5b8c\u6210\u5bf9\u6a2a\u5411\u6781\u9650\u6982\u5ff5\u7684\u7406\u89e3\uff1a <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/continuite-dune-fonction-definie-par-morceaux.webp\" alt=\"\" class=\"wp-image-201\" width=\"429\" height=\"299\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p>\u56fe\u5f62\u8868\u793a\u7684\u51fd\u6570\u70b9 x=-2 \u5904\u7684\u6a2a\u5411\u6781\u9650\u662f\u91cd\u5408\u7684\uff0c\u56e0\u4e3a\u65e0\u8bba\u6211\u4eec\u4ece\u5de6\u4fa7\u8fd8\u662f\u4ece\u53f3\u4fa7\u63a5\u8fd1 x=-2\uff0c\u51fd\u6570\u7684\u503c\u90fd\u8d8b\u5411\u4e8e 3\u3002\u56e0\u6b64\uff0c\u51fd\u6570\u5728 x=-2 \u5904\u7684\u6781\u9650\u7b49\u4e8e 3\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-b7e7a7ee82b827ba469558c38fc81a45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -2^-}f(x)=\\lim_{x\\to -2^+}f(x)=3 \\ \\longrightarrow \\ \\lim_{x\\to -2}f(x)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"389\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u5728\u70b9 x=4 \u5904\uff0c\u6a2a\u5411\u6781\u9650\u4e0d\u540c\uff0c\u56e0\u4e3a\u4ece\u5de6\u4fa7\u51fd\u6570\u63a5\u8fd1 f(x)=3\uff0c\u4f46\u4ece\u53f3\u4fa7\u51fd\u6570\u63a5\u8fd1 f(x)=2\u3002\u56e0\u6b64\u6b64\u65f6\u51fd\u6570\u7684\u6781\u9650\u4e0d\u5b58\u5728\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-a3a52e97ba4cff3c4cd84977fb27db89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 4^-}f(x)=3 \\neq \\lim_{x\\to 4^+}f(x)=2 \\ \\longrightarrow \\ \\cancel{\\exists} \\ \\lim_{x\\to 4}f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"374\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"calculo-de-limites-laterales\"><\/span>\u6a2a\u5411\u9650\u5236\u7684\u8ba1\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u7ed9\u5b9a\u6a2a\u5411\u6781\u9650\u7684\u5b9a\u4e49\uff0c\u6211\u4eec\u5c06\u901a\u8fc7\u6c42\u89e3\u4ee5\u4e0b\u793a\u4f8b\u6765\u4e86\u89e3\u5982\u4f55\u4ee5\u6570\u503c\u65b9\u5f0f\u8ba1\u7b97\u5b83\u4eec\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-42104fdcfcbc6e35486c13774c7288ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 2}\\frac{3}{x-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"74\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u5982\u679c\u6211\u4eec\u7167\u5e38\u8ba1\u7b97\u6781\u9650\uff0c\u6211\u4eec\u5c31\u5f97\u5230\u5b9e\u6570\u9664\u4ee5 0 \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-d98b33a08f3017676e2271bdb0b325d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 2}\\frac{3}{x-2}=\\frac{3}{2-2}=\\frac{3}{0}=\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"220\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u7136\u800c\uff0c\u5728\u8ba1\u7b97\u6a2a\u5411\u6781\u9650\u65f6\uff0c\u6211\u4eec\u6ca1\u6709\u5f97\u5230\u4efb\u4f55\u4e0d\u786e\u5b9a\u6027\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-df0f7477f84abd22a805c7cf70300f16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 2^-}\\frac{3}{x-2}=\\ \\color{red}\\bm{?}\\color{black} \\qquad \\lim_{x\\to 2^+}\\frac{3}{x-2}=\\ \\color{red}\\bm{?}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"378\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p>\u8981\u8ba1\u7b97\u51fd\u6570\u5728 x=2 \u5904\u4ece\u5de6\u4fa7\u5f00\u59cb\u7684\u6a2a\u5411\u6781\u9650\uff0c\u5fc5\u987b\u53d6\u4e00\u4e2a\u5c0f\u4e8e x=2 \u4f46\u975e\u5e38\u63a5\u8fd1\u7684\u6570\u5b57\uff0c\u4f8b\u5982 x=1.999\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-e3cc43d40eb552932818635f1abcd310_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 2^-}\\frac{3}{x-2}=\\frac{3}{\\color{red}\\bm{1,999}\\color{black}-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"252\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5206\u6bcd\u5c06\u662f\u4e00\u4e2a\u975e\u5e38\u5c0f\u7684\u8d1f\u6570\uff0c\u4f46\u751a\u81f3\u4e0d\u662f\u96f6\uff0c\u901a\u5e38\u7528\u96f6\u548c\u524d\u9762\u7684\u51cf\u53f7\u8868\u793a\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-32da1df64469d97474fd1b9e25efcd31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 2^-}\\frac{3}{x-2}=\\frac{3}{1,999-2}=\\frac{3}{\\color{red}\\bm{-0}\\color{black}}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"302\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u6a2a\u5411\u6781\u9650\u7684\u7ed3\u679c\u662f\u8d1f\u65e0\u7a77\u5927\uff0c\u56e0\u4e3a\u4efb\u4f55\u6570\u5b57\u9664\u4ee5 0 \u90fd\u4f1a\u5f97\u5230\u65e0\u7a77\u5927\uff0c\u6b63\u6570\u9664\u4ee5\u8d1f\u6570\u90fd\u4f1a\u5f97\u5230\u8d1f\u6570\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-44072ab37d2d34b0a1cf8655d4b576c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 2^-}\\frac{3}{x-2}=\\frac{3}{1,999-2}=\\frac{3}{-0}=\\color{red}\\bm{-\\infty}\\color{black}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"358\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u5de6\u4fa7\u63a5\u8fd1 x=2 \u7684\u503c\u7684\u51fd\u6570\u56fe\u50cf\u6765\u9a8c\u8bc1\u8be5\u51fd\u6570\u662f\u5426\u63a5\u8fd1\u8d1f\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-97d799f09c2e0890cf3a856bf9c711a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\begin{array}{l}f(1,9)=\\cfrac{3}{1,9-2}=-30\\\\[2ex]f(1,99)=\\cfrac{3}{1,99-2}=-300\\\\[2ex]f(1,999)=\\cfrac{3}{1,999-2}=-3000\\\\[2ex]f(1,9999)=\\cfrac{3}{1,9999-2}=-30000\\\\[2ex]f(1,99999)=\\cfrac{3}{1,99999-2}=-300000\\end{array}\\\\[16ex]\\vdots\\\\[1.5ex] f(2^-)=-\\infty\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"317\" width=\"294\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u540c\u6837\uff0c\u8981\u627e\u5230\u51fd\u6570\u5728\u53f3\u4fa7\u70b9 x=2 \u5904\u7684\u6781\u9650\uff0c\u6211\u4eec\u53ef\u4ee5\u5e94\u7528\u76f8\u540c\u7684\u63a8\u7406\uff1a\u6211\u4eec\u53d6\u4e00\u4e2a\u5927\u4e8e 2 \u4f46\u975e\u5e38\u63a5\u8fd1\u7684\u503c\uff0c\u4f8b\u5982 2001\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-21cbd07eef353c53bdfa09eaceb6bd25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 2^+}\\frac{3}{x-2}=\\frac{3}{2,001-2}=\\frac{3}{+0}=+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"292\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u540c\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u4ece\u53f3\u8fb9\u5f00\u59cb\u8d8a\u6765\u8d8a\u63a5\u8fd1x=2\u7684\u51fd\u6570\u56fe\u50cf\u6765\u9a8c\u8bc1\u51fd\u6570\u662f\u5426\u8d8b\u4e8e\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-e6d448cdad3ac6ba82e749b30d2bcc11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\begin{array}{l}f(2,1)=\\cfrac{3}{2,1-2}=30\\\\[2ex]f(2,01)=\\cfrac{3}{2,01-2}=300\\\\[2ex]f(2,001)=\\cfrac{3}{2,001-2}=3000\\\\[2ex]f(2,0001)=\\cfrac{3}{2,0001-2}=30000\\\\[2ex]f(2,00001)=\\cfrac{3}{2,00001-2}=300000\\end{array}\\\\[16ex]\\vdots\\\\[1.5ex] f(2^+)=+\\infty\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"317\" width=\"280\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u5728\u4e0b\u56fe\u4e2d\uff0c\u60a8\u53ef\u4ee5\u770b\u5230\u6240\u5206\u6790\u7684\u51fd\u6570\u3002\u53ef\u4ee5\u770b\u5230\uff0c\u51fd\u6570\u5728\u5de6\u8fb9\u70b9x=2\u5904\u7684\u6a2a\u5411\u6781\u9650\u662f\u8d1f\u65e0\u7a77\u5927\uff0c\u51fd\u6570\u5728\u53f3\u8fb9\u70b9x=2\u5904\u7684\u6a2a\u5411\u6781\u9650\u662f\u6b63\u65e0\u7a77\u5927\u3002 <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/representation-graphique-dune-fonction-de-proportionnalite-inverse.webp\" alt=\"\" class=\"wp-image-166\" width=\"508\" height=\"514\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-limites-laterales\"><\/span>\u6a2a\u5411\u8fb9\u754c\u95ee\u9898\u5df2\u4fee\u590d<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e601<\/h3>\n<p>\u6c42\u4ee5\u4e0b\u5206\u6bb5\u5b9a\u4e49\u51fd\u6570\u5728\u5b9a\u4e49\u53d1\u751f\u53d8\u5316\u7684\u70b9\uff08x=-2 \u548c x=4\uff09\u5904\u7684\u6a2a\u5411\u6781\u9650\u3002 <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercices-resolus-de-fonctions-definies-par-parties.webp\" alt=\"\" class=\"wp-image-209\" width=\"395\" height=\"307\" srcset=\"\" sizes=\"auto, \"><\/figure>\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\">\u6a2a\u5411\u6781\u9650\u5728 x=-2 \u70b9\u5904\u4e0d\u91cd\u5408\uff0c\u5de6\u4fa7\u51fd\u6570\u8d8b\u4e8e f(x)=5\uff0c\u53e6\u4e00\u65b9\u9762\uff0c\u53f3\u4fa7\u51fd\u6570\u4e3a\u5e38\u6570\u4e14\u503c\u4e3a 3\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-36fbddc8f36db5a92d77dd9e9b3b81ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -2^-}f(x)=5\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"118\" 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-8e153627ce1a29511f502c30bd599a6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -2^+}f(x)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"119\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5f53 x \u63a5\u8fd1 4 \u65f6\uff0c\u8fb9\u9650\u4e5f\u4e0d\u540c\u3002\u5206\u6bb5\u51fd\u6570\u4ece\u5de6\u4fa7\u63a5\u8fd1 3\uff0c\u4f46\u4ece\u53f3\u4fa7\u63a5\u8fd1 -2\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-c590b4da48c82d7a43ef65e89b3e9977_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 4^-}f(x)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"109\" 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-a4d4c175b4f2e5ecfe44fb41fe420820_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 4^+}f(x)=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"121\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e602<\/h3>\n<p>\u5224\u65ad\u4e0b\u5217\u5206\u6bb5\u51fd\u6570\u7684 x \u63a5\u8fd1 3 \u65f6\u6781\u9650\u662f\u5426\u5b58\u5728\uff0c\u5982\u679c\u5b58\u5728\uff0c\u5176\u503c\u662f\u591a\u5c11\u3002 <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/limites-laterales-dune-fonction-par-morceaux.webp\" alt=\"\u5206\u6bb5\u51fd\u6570\u7684\u6a2a\u5411\u6781\u9650\" class=\"wp-image-866\" width=\"337\" height=\"303\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5728\u8fd9\u4e2a\u95ee\u9898\u4e2d\uff0c\u4ece\u5de6\u5230\u53f3\u7684\u70b9 x=3 \u5904\u7684\u6a2a\u5411\u6781\u9650\u662f\u76f8\u540c\u7684\uff0c\u56e0\u4e3a\u65e0\u8bba\u4ece\u5de6\u8fd8\u662f\u4ece\u53f3\u63a5\u8fd1\uff0c\u51fd\u6570\u90fd\u8d8b\u5411\u4e8e\u76f8\u540c\u7684\u503c (f(x)=3) \u3002\u4ed6\u7684\u53f3\u4fa7\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-d8f2aea7db3ddf52e74f09960dd15629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 3^-}f(x)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"109\" 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-6ee9606c4d6e820e4ac3bb5fa5aa3575_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 3^+}f(x)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"108\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u6839\u636e\u6781\u9650\u7684\u6570\u5b66\u5b9a\u4e49\uff0c\u5f53x\u8d8b\u5411\u4e8e3\u65f6\u51fd\u6570\u7684\u6781\u9650\u7b49\u4e8e3\uff0c\u56e0\u4e3a\u540c\u4e00\u70b9\u7684\u4e24\u4e2a\u6a2a\u5411\u6781\u9650\u5728\u8be5\u503c\u5904\u91cd\u5408\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-f82de582de8671680e14cfbf92001010_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 3^-}f(x)=\\lim_{x\\to 3^+}f(x)=3 \\ \\longrightarrow \\ \\lim_{x\\to 3}f(x)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"357\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u867d\u7136x=3\u5904\u51fd\u6570\u7684\u6781\u9650\u662f3\uff0c\u4f46\u5fc5\u987b\u8003\u8651\u5230\u6b64\u65f6\u7684\u51fd\u6570\u4e0d\u662f3\uff0c\u800c\u662ff(3)=7\u3002\u6b63\u5982\u6211\u4eec\u7a0d\u540e\u5c06\u770b\u5230\u7684\uff0c\u8fd9\u610f\u5473\u7740\u8be5\u51fd\u6570\u5728 x=3 \u5904\u4e0d\u8fde\u7eed\uff0c\u800c\u662f\u5177\u6709\u53ef\u907f\u514d\u7684\u4e0d\u8fde\u7eed\u6027\u3002<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e603<\/h3>\n<p>\u8ba1\u7b97\u4ee5\u4e0b\u6709\u7406\u51fd\u6570\u5728\u70b9 x=4 \u5904\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-b5eff7e2ff1b4df8ee2ec49b7db80a18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{-2x+3}{x-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"132\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u4e3a\u4e86\u8ba1\u7b97 x \u4ece\u5de6\u8fb9\u8d8b\u5411\u4e8e 4 \u65f6\u7684\u6781\u9650\uff0c\u6211\u4eec\u53d6\u4e00\u4e2a\u5c0f\u4e8e 4 \u4f46\u975e\u5e38\u63a5\u8fd1\u5b83\u7684\u503c\uff0c\u4f8b\u5982 3,999\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-845fc23ff2883308a9b535c84cd69d83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 4^-}\\frac{-2x+3}{x-4}=\\frac{-2\\cdot 3,999+3}{3,999-4}=\\frac{-4,998}{-0}=+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"385\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u5f53 x \u4ece\u5de6\u4fa7\u63a5\u8fd1 4 \u65f6\uff0c\u6a2a\u5411\u6781\u9650\u662f\u6b63\u65e0\u7a77\u5927\u3002<\/p>\n<p class=\"has-text-align-left\">\u4e3a\u4e86\u89e3\u51b3 x \u4ece\u53f3\u4fa7\u8d8b\u5411\u4e8e 4 \u65f6\u7684\u6781\u9650\uff0c\u6211\u4eec\u4ee5\u5927\u4e8e 4 \u4f46\u975e\u5e38\u63a5\u8fd1 4 \u7684\u503c\u6765\u8ba1\u7b97\u51fd\u6570\uff0c\u4f8b\u5982 4,001\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-ef680d721aae4b9b993bbc65542d5d2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 4^+}\\frac{-2x+3}{x-4}=\\frac{-2\\cdot 4,001+3}{4,001-4}=\\frac{-5,002}{+0}=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"385\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u5f53 x \u4ece\u53f3\u4fa7\u63a5\u8fd1 4 \u65f6\uff0c\u6a2a\u5411\u6781\u9650\u4e3a\u8d1f\u65e0\u7a77\u5927\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>\u6c42\u5728\u70b9 x=2 \u5904\u5b9a\u4e49\u7684\u4ee5\u4e0b\u5206\u6bb5\u51fd\u6570\u7684\u6781\u9650\uff08\u5982\u679c\u5b58\u5728\uff09\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-40d5632016e70b9d9ab8e46e76e0102b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} x^2-3 &amp; \\text{si} &amp;  x \\leq 2 \\\\[2ex]\\displaystyle \\frac{-3x+5}{x-3} &amp; \\text{si} &amp; x>2 \\end{array} \\right.&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;75&#8243; width=&#8221;235&#8243; style=&#8221;vertical-align: 0px;&#8221;><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u95ee\u9898\u9648\u8ff0\u8981\u6c42\u6211\u4eec\u627e\u5230\u5206\u6bb5\u51fd\u6570\u6539\u53d8\u8868\u8fbe\u5f0f\u7684\u6781\u9650\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4f7f\u7528\u7b2c\u4e00\u4e2a\u8868\u8fbe\u5f0f\u627e\u5230\u5de6\u4fa7\u7684\u6781\u9650\uff0c\u4f7f\u7528\u7b2c\u4e8c\u4e2a\u8868\u8fbe\u5f0f\u627e\u5230\u53f3\u4fa7\u7684\u6781\u9650\u3002 <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-312aa6dc645115b9d1a680ef3cc5fb9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 2^-}f(x)=\\lim_{x\\to 2^-}(x^2-3)=2^2-3=1\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"303\" 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-cad9d8ff045408344a1435ec6441aa73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 2^+}f(x)=\\lim_{x\\to 2^+}\\frac{-3x+5}{x-3}=\\frac{-3\\cdot 2+5}{2-3}=\\frac{-1}{-1}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"392\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5de6\u4fa7 x=2 \u5904\u51fd\u6570\u7684\u6781\u9650\u4e0e\u53f3\u4fa7\u51fd\u6570\u7684\u6781\u9650\u91cd\u5408\uff0c\u56e0\u6b64\u8be5\u51fd\u6570\u7684\u6781\u9650\u5b58\u5728\u4e14\u7b49\u4e8e 1\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fe286d19fdb8b3f5859e8073869660ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 2^-}f(x)=\\lim_{x\\to 2^+}f(x)=1 \\ \\longrightarrow \\ \\lim_{x\\to 2}f(x)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"356\" 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\u4ec0\u4e48\u662f\u51fd\u6570\u7684\u6a2a\u5411\u6781\u9650\uff08\u5e76\u9644\u6709\u793a\u4f8b\uff09\u3002\u6211\u4eec\u8fd8\u6559\u60a8\u5982\u4f55\u4ee5\u56fe\u5f62\u548c\u6570\u5b57\u65b9\u5f0f\u8ba1\u7b97\u51fd\u6570\u7684\u5de6\u4fa7\u548c\u53f3\u4fa7\u6781\u9650 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/cn\/%e6%a8%aa%e5%90%91%e9%99%90%e5%88%b6\/\"> <span class=\"screen-reader-text\">\u6a2a\u5411\u9650\u5236<\/span> \u67e5\u770b\u5168\u6587 &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[11],"tags":[],"class_list":["post-364","post","type-post","status-publish","format-standard","hentry","category-11"],"yoast_head":"<!-- This site is optimized 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