{"id":29521,"date":"2024-11-18T02:30:48","date_gmt":"2024-11-18T02:30:48","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29521"},"modified":"2024-11-18T03:04:10","modified_gmt":"2024-11-18T03:04:10","slug":"%e6%b8%90%e8%bf%91%e7%ba%bf%e3%80%81%e6%9e%81%e9%99%90%e5%92%8c%e5%9b%be%e5%bd%a2%e8%a1%a8%e7%a4%ba%e6%8a%80%e6%9c%af","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/%e6%b8%90%e8%bf%91%e7%ba%bf%e3%80%81%e6%9e%81%e9%99%90%e5%92%8c%e5%9b%be%e5%bd%a2%e8%a1%a8%e7%a4%ba%e6%8a%80%e6%9c%af\/","title":{"rendered":"\u6e10\u8fd1\u7ebf\u3001\u6781\u9650\u548c\u56fe\u5f62\u8868\u793a\u6280\u672f"},"content":{"rendered":"<style>\np {\n    text-align: justify;\n}\n<\/style>\n<h1 style=\"text-align: center;\">\u6e10\u8fd1\u7ebf\u3001\u6781\u9650\u4e0e\u56fe\u5f62\u8868\u793a\u6280\u672f<\/h1>\n<p style=\"text-align: center;\"><em><strong>\u6458\u8981\uff1a<\/strong><br \/>\n\u5728\u672c\u8282\u8bfe\u4e2d\uff0c\u6211\u4eec\u8ba8\u8bba\u4e86\u51fd\u6570\u5206\u6790\u4e2d\u7684\u6e10\u8fd1\u7ebf\u548c\u4e3b\u5bfc\u9879\u7684\u6982\u5ff5\u3002\u6211\u4eec\u63a2\u7d22\u4e86\u63cf\u8ff0\u51fd\u6570\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u8d8b\u5411\u65e0\u7a77\u5927\u65f6\u884c\u4e3a\u7684\u6c34\u5e73\u6e10\u8fd1\u7ebf\uff1b\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u63a5\u8fd1\u67d0\u4e9b\u503c\u65f6\u6307\u793a\u65e0\u9650\u6781\u9650\u7684\u5782\u76f4\u6e10\u8fd1\u7ebf\uff1b\u4ee5\u53ca\u5728\u5206\u5b50\u6b21\u6570\u9ad8\u4e8e\u5206\u6bcd\u6b21\u6570\u65f6\u4e0e\u51fd\u6570\u76f8\u5173\u7684\u659c\u6e10\u8fd1\u7ebf\u3002\u6b64\u5916\uff0c\u6211\u4eec\u8fd8\u5206\u6790\u4e86\u4e00\u4e2a\u51fd\u6570\u7684\u4e3b\u5bfc\u9879\uff0c\u5b83\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u53d6\u5927\u503c\u6216\u63a5\u8fd1\u67d0\u4e9b\u70b9\u65f6\u63d0\u4f9b\u4e86\u8fd1\u4f3c\u3002<\/em><\/p>\n<p style=\"text-align: center;\"><strong>\u5b66\u4e60\u76ee\u6807<\/strong><br \/>\n\u5b8c\u6210\u672c\u8bfe\u540e\uff0c\u5b66\u751f\u5e94\u80fd\u591f\uff1a<\/p>\n<ol>\n<li><strong>\u7406\u89e3<\/strong>\u6c34\u5e73\u6e10\u8fd1\u7ebf\u7684\u6982\u5ff5\u53ca\u5176\u5728\u5206\u6790\u51fd\u6570\u8d8b\u5411\u65e0\u7a77\u5927\u65f6\u7684\u5e94\u7528\u3002<\/li>\n<li><strong>\u8bc6\u522b<\/strong>\u5782\u76f4\u6e10\u8fd1\u7ebf\u5b58\u5728\u7684\u6761\u4ef6\uff0c\u5e76\u5c06\u5176\u5e94\u7528\u4e8e\u7814\u7a76\u51fd\u6570\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u63a5\u8fd1\u67d0\u4e9b\u503c\u65f6\u7684\u65e0\u9650\u6781\u9650\u3002<\/li>\n<li><strong>\u5206\u6790<\/strong>\u5728\u5206\u5b50\u6b21\u6570\u9ad8\u4e8e\u5206\u6bcd\u6b21\u6570\u65f6\uff0c\u5206\u5f0f\u51fd\u6570\u4e2d\u7684\u659c\u6e10\u8fd1\u7ebf\u3002<\/li>\n<li><strong>\u5e94\u7528<\/strong>\u4e3b\u5bfc\u9879\u7684\u6982\u5ff5\uff0c\u4ee5\u8fd1\u4f3c\u51fd\u6570\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u53d6\u5927\u503c\u6216\u63a5\u8fd1\u67d0\u4e9b\u70b9\u65f6\u7684\u884c\u4e3a\u3002<\/li>\n<li><strong>\u89e3\u91ca<\/strong>\u6e10\u8fd1\u7ebf\u548c\u4e3b\u5bfc\u9879\u5206\u6790\u5982\u4f55\u6709\u52a9\u4e8e\u7406\u89e3\u51fd\u6570\u7684\u6574\u4f53\u884c\u4e3a\u3002<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><strong><u>\u5185\u5bb9\u76ee\u5f55<\/u>\uff1a<\/strong><br \/>\n<a href=\"#1\">\u4ecb\u7ecd<\/a><br \/>\n<a href=\"#2\">\u6c34\u5e73\u6e10\u8fd1\u7ebf\u4e0e\u65e0\u7a77\u6781\u9650<\/a><br \/>\n<a href=\"#3\">\u5782\u76f4\u6e10\u8fd1\u7ebf\u4e0e\u65e0\u9650\u6781\u9650<\/a><br \/>\n<a href=\"#4\">\u659c\u6e10\u8fd1\u7ebf\u3001\u66f2\u7ebf\u548c\u4e3b\u5bfc\u9879<\/a><br \/>\n<a href=\"#5\">\u89e3\u7b54\u7ec3\u4e60<\/a><br \/>\n<a href=\"#6\">\u7ec3\u4e60\u9898<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/Ekd0oSvMbfE\" title=\"YouTube \u89c6\u9891\u64ad\u653e\u5668\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u4ecb\u7ecd<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=Ekd0oSvMbfE&amp;t=98s\" target=\"_blank\" rel=\"noopener\"><strong>\u6211\u4eec\u5df2\u7ecf\u590d\u4e60\u7684\u6781\u9650<\/strong><\/a> \u4f7f\u6211\u4eec\u80fd\u591f\u5b9a\u4e49\u4e00\u4e9b\u6709\u52a9\u4e8e\u7406\u89e3\u51fd\u6570\u6574\u4f53\u884c\u4e3a\u7684\u6982\u5ff5\uff0c\u8fd9\u4e9b\u6982\u5ff5\u5305\u62ec\u4e3b\u5bfc\u9879\u4ee5\u53ca\u6c34\u5e73\u548c\u5782\u76f4\u6e10\u8fd1\u7ebf\u3002\u8fd9\u4e9b\u662f\u51fd\u6570\u56fe\u50cf\u8d8b\u5411\u4e8e\u9760\u8fd1\u7684\u66f2\u7ebf\uff0c\u968f\u7740 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u8d8b\u5411\u67d0\u4e2a\u503c\uff0c\u53ef\u4ee5\u65e0\u9650\u63a5\u8fd1\u8fd9\u4e9b\u66f2\u7ebf\u3002<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u6c34\u5e73\u6e10\u8fd1\u7ebf\u4e0e\u65e0\u7a77\u6781\u9650<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=Ekd0oSvMbfE&amp;t=137s\" target=\"_blank\" rel=\"noopener\"><strong>\u5982\u679c <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u662f\u5b9a\u4e49\u5728<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">]a,+\\infty[<\/span> \u4e0a\u7684\u67d0\u4e2a\u51fd\u6570\uff0c\u90a3\u4e48\u5c31\u6709\u53ef\u80fd\u8ba1\u7b97 <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u8d8b\u5411\u65e0\u7a77\u5927\u65f6\u7684\u6781\u9650\u3002\u5982\u679c\u8be5\u6781\u9650\u5b58\u5728\uff0c\u90a3\u4e48\u53ef\u4ee5\u901a\u8fc7\u8be5\u6781\u9650\u5b9a\u4e49 <strong>\u5411\u53f3\u7684\u6c34\u5e73\u6e10\u8fd1\u7ebf<\/strong>\uff0c\u5176\u65b9\u7a0b\u4e3a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">A_+(x) = L^+<\/span>\n<p>\u5176\u4e2d<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to+\\infty}f(x) = L^+<\/span>\n<p>\u7c7b\u4f3c\u5730\uff0c\u53ef\u4ee5\u5b9a\u4e49 <strong>\u5411\u5de6\u7684\u6c34\u5e73\u6e10\u8fd1\u7ebf<\/strong>\uff0c\u5176\u65b9\u7a0b\u4e3a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">A_-(x) = L^-<\/span>\n<p>\u5f53<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to-\\infty}f(x) = L^-<\/span>\n<p>\u6c34\u5e73\u6e10\u8fd1\u7ebf\u6709\u52a9\u4e8e\u63cf\u8ff0\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u503c\u8d8b\u5411\u65e0\u7a77\u5927\u65f6\u7684\u884c\u4e3a\u3002<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-ckBGkFWse2w\/YH1GWClIciI\/AAAAAAAAE6s\/zZ_se7yShqMLiEHKNT_jkgAWuK9cme5wwCLcBGAsYHQ\/s0\/as%25C3%25ADntotahorizontal.PNG\" alt=\"\u6c34\u5e73\u6e10\u8fd1\u7ebf\" class=\"aligncenter lazyload\" width=\"478\" height=\"290\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-ckBGkFWse2w\/YH1GWClIciI\/AAAAAAAAE6s\/zZ_se7yShqMLiEHKNT_jkgAWuK9cme5wwCLcBGAsYHQ\/s0\/as%25C3%25ADntotahorizontal.PNG\" alt=\"\u6c34\u5e73\u6e10\u8fd1\u7ebf\" class=\"aligncenter lazyload\" width=\"478\" height=\"290\" \/><\/noscript><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u5782\u76f4\u6e10\u8fd1\u7ebf\u4e0e\u65e0\u9650\u6781\u9650<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=Ekd0oSvMbfE&amp;t=277s\" target=\"_blank\" rel=\"noopener\"><strong>\u4e0e\u6c34\u5e73\u6e10\u8fd1\u7ebf\u7c7b\u4f3c\uff0c<\/strong><\/a>\u53ef\u4ee5\u5b9a\u4e49\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u7684 <strong>\u5411\u4e0a\u7684\u5782\u76f4\u6e10\u8fd1\u7ebf<\/strong>\uff0c\u5176\u65b9\u7a0b\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">x=a<\/span>\uff0c\u5f53<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to a}f(x) = +\\infty<\/span>\n<p>\u5982\u679c\u6ee1\u8db3<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to a}f(x) = -\\infty<\/span>\n<p>\u5219\u4e3a <strong>\u5411\u4e0b\u7684\u5782\u76f4\u6e10\u8fd1\u7ebf<\/strong>\u3002\u6839\u636e\u4fa7\u6781\u9650\u7684\u903b\u8f91\uff0c\u6e10\u8fd1\u7ebf\u53ef\u4ee5\u662f\u4ece\u53f3\u4fa7\u6216\u5de6\u4fa7\u3002<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-ptEipMpyIhc\/YH1VDclyMxI\/AAAAAAAAE60\/LmzpK2HAU7oLpswJQy5_TLIv9jSf9whDwCLcBGAsYHQ\/s0\/asintotavertical.PNG\" alt=\"\u5782\u76f4\u6e10\u8fd1\u7ebf\" class=\"aligncenter lazyload\" width=\"428\" height=\"283\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-ptEipMpyIhc\/YH1VDclyMxI\/AAAAAAAAE60\/LmzpK2HAU7oLpswJQy5_TLIv9jSf9whDwCLcBGAsYHQ\/s0\/asintotavertical.PNG\" alt=\"\u5782\u76f4\u6e10\u8fd1\u7ebf\" class=\"aligncenter lazyload\" width=\"428\" height=\"283\" \/><\/noscript><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u659c\u6e10\u8fd1\u7ebf\u3001\u66f2\u7ebf\u548c\u4e3b\u5bfc\u9879<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=Ekd0oSvMbfE&amp;t=400s\" target=\"_blank\" rel=\"noopener\"><strong>\u6700\u7b80\u5355\u7684\u659c\u6e10\u8fd1\u7ebf\u51fa\u73b0<\/strong><\/a>\u5728\u6211\u4eec\u5904\u7406\u5206\u5f0f\u51fd\u6570\u65f6<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{P(x)}{Q(x)}<\/span>\n<p>\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> \u662f\u591a\u9879\u5f0f\u3002\u5982\u679c <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u7684\u6b21\u6570\u6bd4 <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> \u7684\u6b21\u6570\u9ad8\u51fa\u4e00\uff0c\u90a3\u4e48\u53ef\u4ee5\u901a\u8fc7\u591a\u9879\u5f0f\u9664\u6cd5\u5f97\u5230\u7c7b\u4f3c<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{P(x)}{Q(x)} = C(x) + \\dfrac{r(x)}{Q(x)}<\/span>\n<p>\u7684\u7ed3\u679c\uff0c\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">C(x)<\/span> \u662f\u9664\u6cd5\u7684\u5546\uff0c\u4e14 <span class=\"katex-eq\" data-katex-display=\"false\">r(x)<\/span> \u662f\u4f59\u6570\u3002\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u7684\u6b21\u6570\u6bd4 <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> \u9ad8\u51fa\u4e00\u65f6\uff0c<span class=\"katex-eq\" data-katex-display=\"false\">C(x)<\/span> \u662f\u4e00\u6b21\u51fd\u6570\uff0c\u56e0\u6b64\u79f0 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u6709\u4e00\u4e2a\u659c\u6e10\u8fd1\u7ebf\u3002<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-wRTmSl2Z3HE\/YH1dSl-noDI\/AAAAAAAAE68\/og2lPX_ydUUGlxYnn5hgj2mNCSeAPoQKACLcBGAsYHQ\/s0\/asintotaoblicua.PNG\" alt=\"\u659c\u6e10\u8fd1\u7ebf\" class=\"aligncenter lazyload\" width=\"404\" height=\"239\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-wRTmSl2Z3HE\/YH1dSl-noDI\/AAAAAAAAE68\/og2lPX_ydUUGlxYnn5hgj2mNCSeAPoQKACLcBGAsYHQ\/s0\/asintotaoblicua.PNG\" alt=\"\u659c\u6e10\u8fd1\u7ebf\" class=\"aligncenter lazyload\" width=\"404\" height=\"239\" \/><\/noscript><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=Ekd0oSvMbfE&amp;t=633s\" target=\"_blank\" rel=\"noopener\"><strong>\u4e00\u822c\u6765\u8bf4\uff0c\u5982\u679c <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span><\/strong><\/a>\u7684\u6b21\u6570\u6bd4 <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> \u9ad8\uff0c\u5219 <span class=\"katex-eq\" data-katex-display=\"false\">C(x)<\/span> \u7684\u6b21\u6570\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> \u4e4b\u95f4\u7684\u5dee\u3002\u56e0\u6b64\uff0c\u51fd\u6570\u7684\u884c\u4e3a\u662f\u201c\u6e10\u8fd1\u903c\u8fd1\u201d\u4e8e <span class=\"katex-eq\" data-katex-display=\"false\">C(x)<\/span>\uff0c\u8fd9\u5c31\u662f\u51fd\u6570\u7684\u5927\u503c <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u7684 <strong>\u4e3b\u5bfc\u9879<\/strong>\u3002<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-IEYO071tTuY\/YH1fjR-WWnI\/AAAAAAAAE7E\/ga2rZ02i8QU5R1IMvQB9rpgFuDknAGfbACLcBGAsYHQ\/s0\/terminoDominante.PNG\" alt=\"\u4e3b\u5bfc\u9879\u4e0e\u5782\u76f4\u6e10\u8fd1\u7ebf\" class=\"aligncenter lazyload\" width=\"479\" height=\"437\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-IEYO071tTuY\/YH1fjR-WWnI\/AAAAAAAAE7E\/ga2rZ02i8QU5R1IMvQB9rpgFuDknAGfbACLcBGAsYHQ\/s0\/terminoDominante.PNG\" alt=\"\u4e3b\u5bfc\u9879\u4e0e\u5782\u76f4\u6e10\u8fd1\u7ebf\" class=\"aligncenter lazyload\" width=\"479\" height=\"437\" \/><\/noscript><\/p>\n<p>\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u63a5\u8fd1\u67d0\u4e2a <span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{R}<\/span> \u65f6\uff0c\u4e5f\u53ef\u4ee5\u8ba8\u8bba\u4e3b\u5bfc\u9879\u3002<\/p>\n<p>\u5982\u679c <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = P(x)\/Q(x) = C(x) + r(x)\/Q(x),<\/span> \u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">P(x),<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x),<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">r(x)<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">C(x)<\/span> \u5747\u4e3a\u591a\u9879\u5f0f\uff0c\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to a}f(x) = \\infty<\/span>\uff0c\u5219\u5546 <span class=\"katex-eq\" data-katex-display=\"false\">r(x)\/Q(x)<\/span> \u662f <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x=a<\/span> \u9644\u8fd1\u7684 <strong>\u4e3b\u5bfc\u9879<\/strong>\u3002<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>\u89e3\u7b54\u7ec3\u4e60<\/h2>\n<h3><strong>\u7ec3\u4e60 1\uff1a<\/strong><\/h3>\n<p>\u786e\u5b9a\u51fd\u6570\u7684\u6c34\u5e73\u6e10\u8fd1\u7ebf\u548c\u5782\u76f4\u6e10\u8fd1\u7ebf<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{3x + 1}{x - 2}<\/span>\n<p><strong>\u89e3\u7b54\uff1a<\/strong><\/p>\n<p>\u4e3a\u4e86\u6c42 <strong>\u6c34\u5e73\u6e10\u8fd1\u7ebf<\/strong>\uff0c\u8ba1\u7b97 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x \\to \\pm\\infty<\/span> \u65f6\u7684\u6781\u9650\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x \\to \\pm\\infty} \\dfrac{3x + 1}{x - 2} = 3<\/span>\n<p>\u56e0\u6b64\uff0c\u6c34\u5e73\u6e10\u8fd1\u7ebf\u662f <span class=\"katex-eq\" data-katex-display=\"false\">y = 3<\/span>\u3002<\/p>\n<p>\u5bf9\u4e8e <strong>\u5782\u76f4\u6e10\u8fd1\u7ebf<\/strong>\uff0c\u786e\u5b9a\u4f7f\u5206\u6bcd\u4e3a\u96f6\u7684\u503c\uff0c\u5373 <span class=\"katex-eq\" data-katex-display=\"false\">x = 2<\/span>\u3002<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x \\to 2^\\pm} \\dfrac{3x + 1}{x - 2} = \\pm\\infty<\/span>\n<p>\u8fd9\u8868\u793a\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x = 2<\/span> \u5904\u6709\u5782\u76f4\u6e10\u8fd1\u7ebf\u3002<\/p>\n<p><strong>\u6700\u7ec8\u7ed3\u679c\uff1a<\/strong> \u51fd\u6570\u6709\u4e00\u4e2a\u6c34\u5e73\u6e10\u8fd1\u7ebf <span class=\"katex-eq\" data-katex-display=\"false\">y = 3<\/span> \u548c\u4e00\u4e2a\u5782\u76f4\u6e10\u8fd1\u7ebf <span class=\"katex-eq\" data-katex-display=\"false\">x = 2<\/span>\u3002<\/p>\n<h3><strong>\u7ec3\u4e60 2\uff1a<\/strong><\/h3>\n<p>\u627e\u51fa\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">g(x) = \\frac{2x^2 + 3x + 4}{x + 1}<\/span> \u7684\u6c34\u5e73\u6e10\u8fd1\u7ebf\u548c\u659c\u6e10\u8fd1\u7ebf\uff08\u5982\u679c\u5b58\u5728\uff09\u3002<\/p>\n<p><strong>\u89e3\u7b54\uff1a<\/strong><\/p>\n<p>\u9996\u5148\uff0c\u901a\u8fc7\u8ba1\u7b97 <span class=\"katex-eq\" data-katex-display=\"false\">x \\to \\pm\\infty<\/span> \u65f6\u7684\u6781\u9650\u6765\u67e5\u627e <strong>\u6c34\u5e73\u6e10\u8fd1\u7ebf<\/strong>\u3002\u7531\u4e8e\u5206\u5b50\u7684\u6b21\u6570\u6bd4\u5206\u6bcd\u9ad8\uff0c\u6240\u4ee5\u4e0d\u5b58\u5728\u6c34\u5e73\u6e10\u8fd1\u7ebf\u3002<\/p>\n<p>\u5bf9\u4e8e <strong>\u659c\u6e10\u8fd1\u7ebf<\/strong>\uff0c\u6267\u884c\u591a\u9879\u5f0f\u9664\u6cd5\u5f97\u5230\u4ee5\u4e0b\u7ed3\u679c\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{2x^2 + 3x + 4}{x + 1} = 2x + 1 + \\dfrac{3}{x + 1}<\/span>\n<p>\u56e0\u6b64\uff0c\u659c\u6e10\u8fd1\u7ebf\u662f\u76f4\u7ebf <span class=\"katex-eq\" data-katex-display=\"false\">y = 2x + 1<\/span>\uff0c\u5b83\u662f\u51fd\u6570\u7684\u4e3b\u5bfc\u9879\u3002<\/p>\n<p><strong>\u6700\u7ec8\u7ed3\u679c\uff1a<\/strong> \u8be5\u51fd\u6570\u6ca1\u6709\u6c34\u5e73\u6e10\u8fd1\u7ebf\uff0c\u4f46\u6709\u4e00\u4e2a\u659c\u6e10\u8fd1\u7ebf\uff0c\u5176\u65b9\u7a0b\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">y = 2x + 1<\/span>\u3002<\/p>\n<h3><strong>\u7ec3\u4e60 3\uff1a<\/strong><\/h3>\n<p>\u8ba1\u7b97\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">h(x) = \\frac{5}{x^2 - 4}<\/span> \u7684\u5782\u76f4\u6e10\u8fd1\u7ebf\u3002<\/p>\n<p><strong>\u89e3\u7b54\uff1a<\/strong><\/p>\n<p>\u4e3a\u4e86\u627e\u5230 <strong>\u5782\u76f4\u6e10\u8fd1\u7ebf<\/strong>\uff0c\u786e\u5b9a\u4f7f\u5206\u6bcd\u4e3a\u96f6\u7684\u503c\uff0c\u5373 <span class=\"katex-eq\" data-katex-display=\"false\">x^2 - 4 = 0<\/span>\u3002\u8fd9\u53d1\u751f\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x = \\pm 2<\/span>\u3002<\/p>\n<p>\u8bc4\u4f30\u6bcf\u4e2a\u503c\u7684\u4fa7\u6781\u9650\uff1a<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x \\to 2^\\pm} \\dfrac{5}{x^2 - 4} = \\pm\\infty<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x \\to -2^\\pm} \\dfrac{5}{x^2 - 4} = \\pm\\infty<\/span>\n<p><strong>\u6700\u7ec8\u7ed3\u679c\uff1a<\/strong> \u8be5\u51fd\u6570\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x = 2<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">x = -2<\/span> \u5904\u6709\u5782\u76f4\u6e10\u8fd1\u7ebf\u3002<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>\u7ec3\u4e60\u9898<\/h2>\n<ol>\n<li>\u5206\u6790\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\frac{2x^2 - 3x + 1}{x^2 + x - 2}<\/span>\u3002\u786e\u5b9a\u5176\u6c34\u5e73\u6e10\u8fd1\u7ebf\u3001\u5782\u76f4\u6e10\u8fd1\u7ebf\u548c\u659c\u6e10\u8fd1\u7ebf\uff08\u5982\u679c\u5b58\u5728\uff09\u3002\u89e3\u91ca\u6bcf\u4e00\u6b65\u4ee5\u5f3a\u5316\u6e10\u8fd1\u7ebf\u7684\u6982\u5ff5\u53ca\u6781\u9650\u7684\u8ba1\u7b97\u3002<\/li>\n<li>\u8bc4\u4f30\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">g(x) = \\frac{3x^3 + 2x}{x^2 + 1}<\/span>\u3002\u786e\u5b9a <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u8d8b\u5411\u65e0\u7a77\u5927\u65f6\u7684\u4e3b\u5bfc\u9879\u3002\u7136\u540e\uff0c\u9a8c\u8bc1\u662f\u5426\u5b58\u5728\u659c\u6e10\u8fd1\u7ebf\uff0c\u5e76\u89e3\u91ca\u539f\u56e0\u3002<\/li>\n<li>\u8bbe\u8ba1\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">h(x) = \\frac{5x - 4}{x + 1}<\/span> \u7684\u8fd1\u4f3c\u56fe\u3002\u5305\u62ec\u6c34\u5e73\u3001\u5782\u76f4\u548c\u659c\u6e10\u8fd1\u7ebf\uff08\u5982\u679c\u5b58\u5728\uff09\uff0c\u5e76\u5206\u6790 <span class=\"katex-eq\" data-katex-display=\"false\">h(x)<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u53d6\u6781\u503c\u65f6\u7684\u884c\u4e3a\u3002<\/li>\n<li>\u68c0\u9a8c\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">k(x) = \\frac{x^2 - 4x + 3}{x^2 - 1}<\/span> \u662f\u5426\u6709\u5782\u76f4\u6e10\u8fd1\u7ebf\u3002\u8ba8\u8bba\u4e3b\u5bfc\u9879\u5728\u51fd\u6570\u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\u7684\u6781\u9650\u5206\u6790\u4e2d\u7684\u4f5c\u7528\u3002<\/li>\n<li>\u63a2\u7d22\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">m(x) = \\frac{2x^4 + 3x^2 - x + 5}{x^3 - x^2 + 2}<\/span> \u7684\u4e3b\u5bfc\u9879\u3002\u786e\u5b9a <span class=\"katex-eq\" data-katex-display=\"false\">x \\to \\pm\\infty<\/span> \u65f6 <span class=\"katex-eq\" data-katex-display=\"false\">m(x)<\/span> \u7684\u884c\u4e3a\uff0c\u5e76\u5f97\u51fa\u7ed3\u8bba\uff0c\u662f\u5426\u8d8b\u5411\u4e8e\u591a\u9879\u5f0f\u66f2\u7ebf\u800c\u975e\u76f4\u7ebf\u3002<\/li>\n<li>\u81ea\u9009\u4e00\u4e2a\u6709\u7406\u51fd\u6570\uff0c\u8be6\u7ec6\u63cf\u8ff0\u5982\u4f55\u8ba1\u7b97\u5176\u6c34\u5e73\u3001\u5782\u76f4\u548c\u659c\u6e10\u8fd1\u7ebf\uff0c\u4ee5\u53ca\u4e3b\u5bfc\u9879\u3002\u901a\u8fc7\u56fe\u5f62\u5c55\u793a\u4f60\u7684\u53d1\u73b0\uff0c\u4ee5\u4fbf\u53ef\u89c6\u5316\u6bcf\u79cd\u6e10\u8fd1\u7ebf\u7c7b\u578b\u3002<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u6e10\u8fd1\u7ebf\u3001\u6781\u9650\u4e0e\u56fe\u5f62\u8868\u793a\u6280\u672f \u6458\u8981\uff1a \u5728\u672c\u8282\u8bfe\u4e2d\uff0c\u6211\u4eec\u8ba8\u8bba\u4e86\u51fd\u6570\u5206\u6790\u4e2d\u7684\u6e10\u8fd1\u7ebf\u548c\u4e3b\u5bfc\u9879\u7684\u6982\u5ff5\u3002\u6211\u4eec\u63a2\u7d22\u4e86\u63cf\u8ff0\u51fd\u6570\u5728 \u8d8b\u5411\u65e0\u7a77\u5927\u65f6\u884c\u4e3a\u7684\u6c34\u5e73\u6e10\u8fd1\u7ebf\uff1b\u5f53 \u63a5\u8fd1\u67d0\u4e9b\u503c\u65f6\u6307\u793a\u65e0\u9650\u6781\u9650\u7684\u5782\u76f4\u6e10\u8fd1\u7ebf\uff1b\u4ee5\u53ca\u5728\u5206\u5b50\u6b21\u6570\u9ad8\u4e8e\u5206\u6bcd\u6b21\u6570\u65f6\u4e0e\u51fd\u6570\u76f8\u5173\u7684\u659c\u6e10\u8fd1\u7ebf\u3002\u6b64\u5916\uff0c\u6211\u4eec\u8fd8\u5206\u6790\u4e86\u4e00\u4e2a\u51fd\u6570\u7684\u4e3b\u5bfc\u9879\uff0c\u5b83\u5728 \u53d6\u5927\u503c\u6216\u63a5\u8fd1\u67d0\u4e9b\u70b9\u65f6\u63d0\u4f9b\u4e86\u8fd1\u4f3c\u3002 \u5b66\u4e60\u76ee\u6807 \u5b8c\u6210\u672c\u8bfe\u540e\uff0c\u5b66\u751f\u5e94\u80fd\u591f\uff1a \u7406\u89e3\u6c34\u5e73\u6e10\u8fd1\u7ebf\u7684\u6982\u5ff5\u53ca\u5176\u5728\u5206\u6790\u51fd\u6570\u8d8b\u5411\u65e0\u7a77\u5927\u65f6\u7684\u5e94\u7528\u3002 \u8bc6\u522b\u5782\u76f4\u6e10\u8fd1\u7ebf\u5b58\u5728\u7684\u6761\u4ef6\uff0c\u5e76\u5c06\u5176\u5e94\u7528\u4e8e\u7814\u7a76\u51fd\u6570\u5728 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