{"id":27819,"date":"2024-08-11T13:00:06","date_gmt":"2024-08-11T13:00:06","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27819"},"modified":"2024-09-12T19:04:41","modified_gmt":"2024-09-12T19:04:41","slug":"%e5%8d%95%e5%8f%98%e9%87%8f%e5%87%bd%e6%95%b0%e7%9a%84%e6%9e%81%e9%99%90","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/zh\/%e5%8d%95%e5%8f%98%e9%87%8f%e5%87%bd%e6%95%b0%e7%9a%84%e6%9e%81%e9%99%90\/","title":{"rendered":"\u5b9e\u53d8\u91cf\u51fd\u6570\u7684\u6781\u9650"},"content":{"rendered":"<p><center><\/p>\n<h1>\u4e00\u5143\u5b9e\u53d8\u91cf\u51fd\u6570\u7684\u6781\u9650<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center\"><em><strong>\u6458\u8981\uff1a<\/strong><br \/>\n\u672c\u8282\u8bfe\u6df1\u5165\u8ba8\u8bba\u4e86\u4e00\u5143\u5b9e\u53d8\u91cf\u51fd\u6570\u7684\u6781\u9650\u7684\u6b63\u5f0f\u5b9a\u4e49\uff0c\u5e76\u5728\u6b64\u57fa\u7840\u4e0a\u8bc1\u660e\u4e86\u6781\u9650\u4ee3\u6570\u7684\u4e3b\u8981\u6027\u8d28\u3002<\/br><\/em><\/p>\n<p style=\"text-align:center\"><em><strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong><br \/>\n\u672c\u8282\u8bfe\u7ed3\u675f\u65f6\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a<\/p>\n<ul>\n<li><strong>\u8bb0\u4f4f<\/strong>\u4e00\u5143\u5b9e\u53d8\u91cf\u51fd\u6570\u7684\u6781\u9650\u5b9a\u4e49\u3002<\/li>\n<li><strong>\u901a\u8fc7<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon-\\delta<\/span><\/span>\u63a8\u5bfc\u8bc1\u660e<\/strong>\u5bfc\u81f4\u6781\u9650\u4ee3\u6570\u7684\u6027\u8d28\u3002<\/li>\n<li><strong>\u4f7f\u7528\u6781\u9650\u4ee3\u6570\u53ca\u5176\u6027\u8d28<\/strong>\u8ba1\u7b97\u4e00\u5143\u5b9e\u53d8\u91cf\u51fd\u6570\u7684\u6781\u9650\u3002<\/li>\n<\/ul>\n<p><\/em><\/p>\n<p><center><br \/>\n<strong>\u76ee\u5f55<\/strong><br \/>\n<a href=\"#1\"><strong>\u4ecb\u7ecd<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>\u4ece\u56fe\u5f62\u89d2\u5ea6\u770b\u51fd\u6570\u6781\u9650\u7684\u76f4\u89c2\u6982\u5ff5<\/strong><\/a><br \/>\n<a href=\"#3\"><strong>\u6781\u9650\u7684\u6b63\u5f0f\u5b9a\u4e49<\/strong><\/a><br \/>\n<a href=\"#4\"><strong>\u6781\u9650\u7684\u6027\u8d28<\/strong><\/a><br \/>\n<a href=\"#5\">\u5982\u679c\u6781\u9650\u5b58\u5728\uff0c\u90a3\u4e48\u5b83\u662f\u552f\u4e00\u7684<\/a><br \/>\n<a href=\"#6\">\u6781\u9650\u4ee3\u6570<\/a><br \/>\n<a href=\"#7\">\u7b80\u5355\u6781\u9650\u8ba1\u7b97<\/a><br \/>\n<\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/FEPfoAfPsFY\" 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 style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=240s\" rel=\"noopener\" target=\"_blank\"><strong>\u4ee3\u6570\u548c\u51e0\u4f55\u4e0e\u5fae\u79ef\u5206\u7684\u5b66\u4e60\u6709\u4ec0\u4e48\u4e0d\u540c\uff1f<\/strong><\/a> \u8fd9\u4e2a\u95ee\u9898\u7684\u7b54\u6848\u7531\u6781\u9650\u7684\u6982\u5ff5\u7ed9\u51fa\u3002\u56e0\u6b64\uff0c\u672c\u6587\u7ae0\u8ba8\u8bba\u4e86\u6781\u9650\u53ca\u5176\u5b9a\u4e49\u3002<\/p>\n<p style=\"text-align: justify;\">\u6211\u4eec\u901a\u5e38\u5c06\u201c\u6781\u9650\u201d\u4e00\u8bcd\u4e0e\u67d0\u79cd\u8fb9\u754c\u8054\u7cfb\u8d77\u6765\uff0c\u5c31\u50cf\u6709\u7aef\u70b9a\u548cb\u7684\u533a\u95f4\u7684\u8fb9\u754c\uff08\u65e0\u8bba\u5176\u6027\u8d28\u5982\u4f55\uff09\u3002<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b[\\;\\; ;\\;\\; ]a,b]\\;\\; ; \\;\\; ]a,b[\\;\\; ; [a,b] <\/span><\/span>,<\/p>\n<p style=\"text-align: justify;\">\u6216\u8005\u50cf\u201c\u73b0\u5728\u201d\uff0c\u6211\u4eec\u53ef\u4ee5\u8bf4\u5b83\u662f\u8fc7\u53bb\u548c\u672a\u6765\u7684\u8fb9\u754c\u3002\u7c7b\u4f3c\u5730\uff0c\u6781\u9650\u7684\u6982\u5ff5\u5f15\u5165\u4e86\u5bf9\u8fd9\u4e00\u76f4\u89c2\u60f3\u6cd5\u7684\u6570\u5b66\u7406\u89e3\uff0c\u5373\u6e10\u8fd1\u5730\u63a5\u8fd1\u67d0\u4e2a\u70b9\u3002<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u4ece\u56fe\u5f62\u89d2\u5ea6\u770b\u51fd\u6570\u6781\u9650\u7684\u76f4\u89c2\u6982\u5ff5<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=314s\" rel=\"noopener\" target=\"_blank\"><strong>\u4e3a\u4e86\u5f00\u59cb\u76f4\u89c2\u5730\u7406\u89e3\u6781\u9650\u7684\u6982\u5ff5\uff0c\u6700\u597d\u4ece\u51fd\u6570\u7684\u56fe\u5f62\u8868\u793a\u5165\u624b<\/strong><\/a>\uff0c\u5e76\u8be2\u95ee\u5f53 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u968f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u63a5\u8fd1 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u65f6\u5c06\u4f1a\u53d1\u751f\u4ec0\u4e48\u3002<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-rMjBnCIK8Ts\/YGDfXYswS4I\/AAAAAAAAEwA\/1GY0wy3JkXk99kveDTp1SltJOTAITgN3wCLcBGAsYHQ\/s0\/limite.PNG\" alt=\"\u51fd\u6570\u7684\u6781\u9650\" class=\"alignnone size-full lazyload\" width=\"692\" height=\"565\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-rMjBnCIK8Ts\/YGDfXYswS4I\/AAAAAAAAEwA\/1GY0wy3JkXk99kveDTp1SltJOTAITgN3wCLcBGAsYHQ\/s0\/limite.PNG\" alt=\"\u51fd\u6570\u7684\u6781\u9650\" class=\"alignnone size-full lazyload\" width=\"692\" height=\"565\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u63a5\u8fd1 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>\uff0c\u90a3\u4e48\u5c06\u5b58\u5728\u4ee5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u4e3a\u4e2d\u5fc3\uff0c\u534a\u5f84\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u7684\u5f00\u533a\u95f4\uff0c\u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u5305\u542b\u5728\u5176\u4e2d\u3002\u6211\u4eec\u53ef\u4ee5\u7528\u4e09\u79cd\u4e0d\u540c\u7684\u65b9\u5f0f\u6765\u8868\u793a\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|x-x_0|\\lt \\delta<\/span><\/span>,<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|x\\in]x_0 - \\delta , x_0 + \\delta[ <\/span><\/span>,<\/p>\n<p style=\"text-align: center;\">\u6216\u8005 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in\\mathcal{B}(x_0,\\delta)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><em>\u5728\u6211\u4eec\u7684\u4e0a\u4e0b\u6587\u4e2d\uff0c\u8fd9\u4e09\u79cd\u8868\u793a\u65b9\u5f0f\u90fd\u662f\u76f8\u540c\u7684\uff1b\u4f46\u6700\u540e\u4e00\u79cd\uff0c\u5373\u201c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u5305\u542b\u5728\u4ee5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> \u4e3a\u4e2d\u5fc3\uff0c\u534a\u5f84\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u7684\u5f00\u7403\u4e2d\u201d\uff0c\u66f4\u9002\u5408\u7528\u4e8e <strong>\u62d3\u6251\u5b66\u8bfe\u7a0b<\/strong>\uff0c\u5728\u90a3\u91cc\u4f1a\u5bf9\u8fd9\u4e2a\u201c\u90bb\u8fd1\u95ee\u9898\u201d\u8fdb\u884c\u66f4\u6df1\u5165\u7684\u63a2\u8ba8\u3002<\/em><\/p>\n<p style=\"text-align: justify;\">\u5982\u679c\u53d1\u751f\u8fd9\u79cd\u60c5\u51b5\uff0c\u90a3\u4e48\u6211\u4eec\u5c06\u770b\u5230\u5b58\u5728\u53e6\u4e00\u4e2a\u4ee5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> \u4e3a\u4e2d\u5fc3\uff0c\u534a\u5f84\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span> \u7684\u5f00\u533a\u95f4\uff0c\u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u5305\u542b\u5728\u5176\u4e2d\uff0c\u5373\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - l|\\lt \\epsilon<\/span><\/span>\u3002<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-26xU-o1y-Eg\/YGDfXYgOp2I\/AAAAAAAAEwE\/FGMGEQdvRzg_OvnUqKolJ9v51xUVF4O7QCLcBGAsYHQ\/s0\/limite2.PNG\" alt=\"\u51fd\u6570\u7684\u6781\u9650\" class=\"alignnone size-full lazyload\" width=\"625\" height=\"549\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-26xU-o1y-Eg\/YGDfXYgOp2I\/AAAAAAAAEwE\/FGMGEQdvRzg_OvnUqKolJ9v51xUVF4O7QCLcBGAsYHQ\/s0\/limite2.PNG\" alt=\"\u51fd\u6570\u7684\u6781\u9650\" class=\"alignnone size-full lazyload\" width=\"625\" height=\"549\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">\u8fd9\u91cc\u4ea7\u751f\u4e86\u6781\u9650\u6982\u5ff5\u7684\u57fa\u672c\u601d\u60f3\uff0c\u5373\u5f53 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt|x-x_0|\\lt \\delta<\/span><\/span> \u65f6\uff0c\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x)-l|\\lt \\epsilon<\/span><\/span>\uff0c\u5219\u6781\u9650 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> \u5b58\u5728\u3002<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>\u6781\u9650\u7684\u6b63\u5f0f\u5b9a\u4e49<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=689s\" rel=\"noopener\" target=\"_blank\"><strong>\u4ece\u521a\u624d\u5448\u73b0\u7684\u76f4\u89c2\u548c\u56fe\u5f62\u6982\u5ff5\u5f00\u59cb\uff0c\u53ef\u4ee5\u63a8\u5bfc\u51fa\u6781\u9650\u7684\u6b63\u5f0f\u5b9a\u4e49\u3002<\/strong><\/a>\u6211\u4eec\u8bf4\u6781\u9650\u5b58\u5728\uff0c\u5f53\u4e14\u4ec5\u5f53\u65e0\u8bba <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span> \u5982\u4f55\uff08\u5373 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> \u4e4b\u95f4\u7684\u8ddd\u79bb\uff09\uff0c\u603b\u80fd\u627e\u5230\u4e00\u4e2a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span>\uff0c\u4f7f\u5f97\u5f53 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt|x-x_0|\\lt \\delta<\/span><\/span> \u65f6\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - l|\\lt \\epsilon<\/span><\/span>\u3002\u8fd9\u4e2a\u6982\u5ff5\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u8868\u8fbe\u5f0f\u6982\u62ec\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x)=l := \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow |f(x) - l|\\lt \\epsilon\\right)<\/span><\/span>,<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u6781\u9650\u7684\u6027\u8d28<\/h2>\n<p style=\"text-align: justify;\">\u62e5\u6709\u6781\u9650\u7684\u6b63\u5f0f\u5b9a\u4e49\u7684\u597d\u5904\u5728\u4e8e\uff0c\u6211\u4eec\u73b0\u5728\u53ef\u4ee5\u57fa\u4e8e\u8fd9\u4e2a\u5b9a\u4e49\u8bc1\u660e\u6781\u9650\u7684\u5404\u79cd\u6027\u8d28\uff0c\u5305\u62ec\u90a3\u4e9b\u770b\u4f3c\u76f4\u89c2\u548c\u4e0d\u90a3\u4e48\u76f4\u89c2\u7684\u6027\u8d28\u3002<\/p>\n<p style=\"text-align: justify;\">\u5728\u7ee7\u7eed\u4e4b\u524d\uff0c\u867d\u7136\u8fd9\u4e0d\u662f\u4e25\u683c\u5fc5\u8981\u7684\uff0c\u4f46\u6211\u4eec\u5f3a\u70c8\u5efa\u8bae\u4f60\u590d\u4e60\u4e00\u4e9b <a href=\"http:\/\/toposuranos.com\/material\/es\/category\/matematica\/logica-matematica\/logica-proposicional\/\" rel=\"noopener\" target=\"_blank\"><strong>\u6570\u5b66\u903b\u8f91<\/strong><\/a> \u7684\u6982\u5ff5\uff0c\u8fd9\u6837\u4f60\u4f1a\u66f4\u5bb9\u6613\u7406\u89e3\u63a5\u4e0b\u6765\u7684\u8bc1\u660e\u3002<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>\u5982\u679c\u6781\u9650\u5b58\u5728\uff0c\u90a3\u4e48\u5b83\u662f\u552f\u4e00\u7684<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=904s\" rel=\"noopener\" target=\"_blank\"><strong>\u4e3a\u4e86\u8bc1\u660e\u8fd9\u4e00\u6027\u8d28\uff0c\u6211\u4eec\u5c06\u4f7f\u7528\u53cd\u8bc1\u6cd5\u3002<\/strong><\/a>\u9996\u5148\u5b9a\u4e49\u4ee5\u4e0b\u5047\u8bbe\u96c6\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\mathcal{H}= \\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime, L\\neq L^\\prime\\}<\/span><\/span>.<\/p>\n<p style=\"text-align: justify;\">\u57fa\u4e8e\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u5efa\u4ee5\u4e0b\u5f62\u5f0f\u8bc1\u660e\uff1a<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}f(x) = L <\/span><\/span>; <strong>\u5047\u8bbe<\/strong><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow |f(x) - L|\\lt \\epsilon\\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}f(x) = L^\\prime <\/span><\/span>; \u5047\u8bbe<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow |f(x) - L^\\prime |\\lt \\epsilon\\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash L \\neq L^\\prime <\/span><\/span>; \u5047\u8bbe<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow\\right.<\/span><\/span> <span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left. \\left[ \\left( |f(x) - L |\\lt \\epsilon \\right) \\wedge \\left( |f(x) - L^\\prime |\\lt \\epsilon\\right) \\right] \\right. <\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">)<\/span><\/span>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>&#8211;<strong>\u5408\u53d6<\/strong>(1,2)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow\\right.<\/span><\/span> <span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left. \\left[ \\left( |f(x) - L |\\lt \\epsilon \\right) \\wedge \\left( |f(x) - L^\\prime |\\lt \\epsilon\\right) \\right] \\right. <\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">)<\/span><\/span>; <strong>\u5355\u8c03\u6027<\/strong>(4)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash \\epsilon = \\frac{L - L^\\prime}{2}\\gt 0 <\/span><\/span>; \u56e0\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">L \\lt L^\\prime <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash \\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow\\right.<\/span><\/span> <span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left. \\left[ \\left( |f(x) - L |\\lt \\frac{L - L^\\prime}{2} \\right) \\wedge \\left( |f(x) - L^\\prime |\\lt \\frac{L - L^\\prime}{2}\\right) \\right] \\right. <\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">)<\/span><\/span>; \u4f7f\u7528(5,6)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( 2 |f(x) - L |\\lt L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( 2|f(x) - L^\\prime |\\lt L - L^\\prime)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2 (f(x) - L )\\lt L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2(f(x) - L^\\prime )\\lt L - L^\\prime)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2f(x) - 2L \\lt L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2f(x) - 2L^\\prime \\lt L - L^\\prime)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( L + L^\\prime \\lt 2f(x) \\lt 3L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + 3L^\\prime \\lt 2f(x) \\lt L + L^\\prime)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + 3L^\\prime \\lt 2f(x) \\lt L + L^\\prime)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( L + L^\\prime \\lt 2f(x) \\lt 3L - L^\\prime )<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash \\bot <\/span><\/span>; \u7531(1,2,6,7)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(9)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\gt L^\\prime\\}\\vdash \\bot <\/span><\/span>; \u540c\u6837\u7684\u8fc7\u7a0b(8)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(10)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash [(L\\lt L^\\prime) \\vee (L\\gt L^\\prime)] \\rightarrow \\bot <\/span><\/span>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-\u5f15\u5165(8,9)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(11)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash [L\\ \\neq L^\\prime] \\rightarrow \\bot <\/span><\/span>; \u5b9a\u4e49(10)<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(12)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\bot <\/span><\/span>; <strong>MP<\/strong>(3,11)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime, L\\neq L^\\prime\\right\\} \\vdash \\bot <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(13)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime \\right\\} \\vdash \\neg(L\\neq L^\\prime) <\/span><\/span>; <strong>\u77db\u76fe<\/strong>(12)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime \\right\\} \\vdash L = L^\\prime.<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">\u901a\u8fc7\u6b64\u8bc1\u660e\uff0c\u6211\u4eec\u5f97\u51fa\uff0c\u5982\u679c\u5b58\u5728\u4e24\u4e2a\u6781\u9650\uff0c\u5219\u5b83\u4eec\u76f8\u7b49\uff0c\u56e0\u6b64\u6781\u9650\u662f\u552f\u4e00\u7684\u3002<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>\u6781\u9650\u7684\u4ee3\u6570<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=2011s\" rel=\"noopener\" target=\"_blank\"><strong>\u901a\u8fc7\u6211\u4eec\u76ee\u524d\u7684\u5b66\u4e60\uff0c\u6211\u4eec\u5df2\u7ecf\u4e86\u89e3\u4e86\u6781\u9650\u7684\u6570\u5b66\u6982\u5ff5\u7684\u57fa\u672c\u601d\u60f3\u3002<\/strong><\/a> \u4f46\u662f\u4ec5\u4ec5\u4f9d\u9760\u8fd9\u4e9b\u662f\u8fdc\u8fdc\u4e0d\u591f\u7684\uff0c\u8981\u8fdb\u884c\u6781\u9650\u7684\u8ba1\u7b97\uff0c\u7edd\u4e0d\u4f1a\u6709\u4eba\u4ec5\u4ec5\u4f9d\u9760\u6781\u9650\u7684\u5b9a\u4e49\u6765\u8fdb\u884c\u8ba1\u7b97\u3002\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u6211\u4eec\u5c06\u4ecb\u7ecd\u4e00\u4e9b\u6280\u672f\uff0c\u5e2e\u52a9\u6211\u4eec\u5f00\u59cb\u8ba1\u7b97\u4e00\u4e9b\u6781\u9650\u3002<\/p>\n<p style=\"text-align: justify;\">\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0, \\alpha, \\beta, L, M \\in \\mathbb{R},<\/span><\/span> \u4ee5\u53ca f \u548c g \u4e3a\u5b9e\u51fd\u6570\uff0c\u5e76\u4e14\u6ee1\u8db3\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} f(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} g(x) = M<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u90a3\u4e48\uff0c\u4ee5\u4e0b\u6027\u8d28\u6210\u7acb\uff1a<\/p>\n<h4>\u51fd\u6570\u548c\u4e0e\u5dee\u7684\u6781\u9650<\/h4>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\left(\\alpha f(x) \\pm \\beta g(x) \\right) = \\alpha L \\pm \\beta M<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u8bc1\u660e\uff1a<\/strong><\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=2053s\" rel=\"noopener\" target=\"_blank\"><strong>\u8003\u8651\u5047\u8bbe\u96c6<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\mathcal{H}=\\left\\{\\lim_{x\\to x_0} f(x) = L, \\lim_{x\\to x_0} g(x) = M \\right\\}<\/span><\/span>\uff0c\u57fa\u4e8e\u6b64\u6211\u4eec\u53ef\u4ee5\u8fdb\u884c\u4ee5\u4e0b\u63a8\u7406\uff1a<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}f(x) = L <\/span><\/span>; \u5047\u8bbe<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |f(x) - L|\\lt \\epsilon \\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\alpha||f(x) - L|\\lt |\\alpha|\\epsilon \\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left( 0 \\lt|x-x_0|\\lt \\delta \\rightarrow |\\alpha f(x) - \\alpha L|\\lt |\\alpha|\\epsilon \\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\overline{\\epsilon}:= |\\alpha|\\epsilon <\/span><\/span>; \u5b9a\u4e49<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\epsilon} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\alpha f(x) - \\alpha L|\\lt \\overline{\\epsilon} \\right) <\/span><\/span>; \u7531(1,2)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}\\alpha f(x) = \\alpha L <\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}g(x) = M <\/span><\/span>; \u5047\u8bbe<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}\\beta g(x) = \\beta M <\/span><\/span><\/span>; \u4e0e(3)\u7c7b\u4f3c<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\overline{\\epsilon}} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left( 0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\beta g(x) - \\beta M|\\lt \\overline{\\overline{\\epsilon}} \\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\epsilon},\\overline{\\overline{\\epsilon}} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow \\left[|\\alpha f(x) - \\alpha L|+ |\\beta g(x) - \\beta M|\\lt \\overline{\\epsilon}+ \\overline{\\overline{\\epsilon}} \\right] \\right) <\/span><\/span>; \u7531(3,5)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |\\alpha f(x) - \\alpha L + \\beta g(x) - \\beta M| \\leq |\\alpha f(x) - \\alpha L|+ |\\beta g(x) - \\beta M| <\/span><\/span>; \u4e09\u89d2\u4e0d\u7b49\u5f0f: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x,y\\in\\mathbb{R})(|x+y|\\leq |x|+|y|)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\epsilon},\\overline{\\overline{\\epsilon}} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\alpha f(x) - \\alpha L + \\beta g(x) - \\beta M| \\lt \\overline{\\epsilon}+ \\overline{\\overline{\\epsilon}} \\right) <\/span><\/span>; \u7531(6,7)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(9)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon^* := \\overline{\\epsilon} + \\overline{\\overline{\\epsilon}}<\/span><\/span>; \u5b9a\u4e49<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(10)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon^* \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\alpha f(x) + \\beta g(x) - \\alpha L - \\beta M| \\lt \\epsilon^* \\right) <\/span><\/span>; \u7531(8,9)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} (\\alpha f(x) + \\beta g(x)) = \\alpha L + \\beta M <\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(11)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma:= - \\beta<\/span><\/span>; \u5b9a\u4e49<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(12)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} (\\alpha f(x) + \\gamma g(x)) = \\alpha L + \\gamma M <\/span><\/span>; \u7c7b\u4f3c(10)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(13)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span style=\"background-color: #ffff80; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} (\\alpha f(x) - \\beta g(x)) = \\alpha L - \\beta M <\/span><\/span><\/span>; \u7531(11,12)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(14)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} (\\alpha f(x) \\pm \\beta g(x)) = \\alpha L \\pm \\beta M <\/span><\/span>; \u7531(10,13) <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>\u51fd\u6570\u4e58\u79ef\u7684\u6781\u9650<\/h4>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\left( f(x) g(x) \\right) = L M<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=2611s\" rel=\"noopener\" target=\"_blank\"><strong>\u8fd9\u4e2a\u8bc1\u660e\u6bd4\u524d\u4e00\u4e2a\u7a0d\u5fae\u590d\u6742\u4e00\u70b9\uff0c<\/strong><\/a> \u4f46\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4e00\u4e9b\u5c0f\u6280\u5de7\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u3002\u4f7f\u7528\u4e0e\u524d\u9762\u8bc1\u660e\u76f8\u540c\u7684\u5047\u8bbe\u96c6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{H}<\/span><\/span>\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u5efa\u4ee5\u4e0b\u63a8\u7406\uff1a<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\overline{\\epsilon} := \\frac{|\\epsilon|}{2(|M|+1)} \\leq \\frac{|\\epsilon|}{2} <\/span><\/span>; \u5b9a\u4e49<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} f(x) = L <\/span><\/span>; \u5047\u8bbe<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\epsilon} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right)\\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |f(x) - L| \\lt \\overline{\\epsilon} = \\frac{|\\epsilon|}{2(|M|+1)}\\right) <\/span><\/span>; \u4f7f\u7528(1)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\overline{\\overline{\\epsilon}} := \\frac{|\\epsilon|}{2(|L|+1)} \\leq \\frac{|\\epsilon|}{2}<\/span><\/span>; \u5b9a\u4e49<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} g(x) = M <\/span><\/span>; \u5047\u8bbe<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\overline{\\epsilon}} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right)\\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |g(x) - M| \\lt \\overline{\\overline{\\epsilon}} = \\frac{|\\epsilon|}{2(|L|+1)}\\right) <\/span><\/span>; \u4f7f\u7528(3)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)| - |L| \\lt<\/span><\/span> <span style=\"background-color: #a0ffff; color:#000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - L| \\lt \\overline{\\epsilon} \\lt 1 <\/span><\/span><\/span>; \u4e09\u89d2\u4e0d\u7b49\u5f0f + \u7279\u6b8a\u60c5\u51b5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\epsilon}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)|\\lt 1 + |L| <\/span><\/span>; \u7531(5)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |g(x)| - |M| \\lt |g(x) - M| \\lt \\overline{\\overline{\\epsilon}} \\lt 1 <\/span><\/span>; \u4e09\u89d2\u4e0d\u7b49\u5f0f + \u7279\u6b8a\u60c5\u51b5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\overline{\\epsilon}}<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |g(x)| \\lt 1 + |M| <\/span><\/span>; \u7531(7)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(9)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|=|<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)g(x) - Mf(x)<\/span><\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">+ Mf(x) - LM<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|<\/span><\/span>; \u52a0\u96f6<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|=|<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)(g(x) - M)<\/span><\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">+ M (f(x) - L)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|<\/span><\/span>; \u56e0\u5f0f\u5206\u89e3<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(10)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|\\leq |<\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)(g(x) - M)<\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">| + |<\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M (f(x) - L)<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|<\/span><\/span>; \u4e09\u89d2\u4e0d\u7b49\u5f0f(9)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|\\leq <\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x)||g(x) - M|<\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> + <\/span><\/span> <span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|M| |f(x) - L|<\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(11)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|\\lt <\/span><\/span> <span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1 + |L|)|g(x) - M|<\/span><\/span><\/span><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">+<\/span><\/span> <span style=\"background-color: #a0ffff;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|M|\\overline{\\epsilon}<\/span><\/span><\/span>; \u7531(5,6,10)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(12)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left[ |g(x) - M|\\lt \\overline{\\overline{\\epsilon}} \\right] \\rightarrow \\left[ (1+|L|)|g(x) - M| + |M|\\overline{\\epsilon} \\lt (1+|L|)\\overline{\\overline{\\epsilon}} + |M|\\overline{\\epsilon}\\right]<\/span><\/span>; \u7531(11)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(13)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left[ |g(x) - M|\\lt \\overline{\\overline{\\epsilon}} \\right] \\rightarrow \\left[ (1+|L|)|g(x) - M| + |M|\\overline{\\epsilon} \\lt (1+|L|)\\frac{|\\epsilon|}{2(|L|+1)} + |M|\\frac{|\\epsilon|}{2(|M|+1)}\\right]<\/span><\/span>; \u7531(1,3,12)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left[ |g(x) - M|\\lt \\overline{\\overline{\\epsilon}} \\right] \\rightarrow \\left[ (1+|L|)|g(x) - M| + |M|\\overline{\\epsilon} \\lt \\frac{|\\epsilon|}{2} + \\frac{|\\epsilon||M|}{2(|M|+1)} \\lt \\frac{|\\epsilon|}{2}+ \\frac{|\\epsilon|}{2} = |\\epsilon| \\right]<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(14)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left[ |g(x) - M|\\lt \\overline{\\overline{\\epsilon}} \\right] \\rightarrow \\left[ |f(x)g(x) - LM|\\lt |\\epsilon| \\right]<\/span><\/span>; \u7531(11,13)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(15)<\/span><\/span><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash (\\forall \\epsilon \\gt 0 ) (\\exists \\delta \\gt 0 ) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |f(x)g(x) - LM|\\lt |\\epsilon| \\leq \\epsilon \\right) <\/span><\/span>; \u7531(1,2,4,14)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}f(x)g(x) = LM.<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>\u5e38\u6570\u51fd\u6570\u7684\u6781\u9650<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3450s\" rel=\"noopener\" target=\"_blank\"><strong>\u5e38\u6570\u51fd\u6570\u7684\u6781\u9650<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=c<\/span><\/span>, \u662f\u5e38\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span>\u3002\u4e5f\u5c31\u662f\u8bf4\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}c = c<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u8bc1\u660e<\/strong><\/p>\n<p style=\"text-align: justify;\">\u8fd9\u4e2a\u8bc1\u660e\u5176\u5b9e\u5f88\u7b80\u5355\uff0c\u56e0\u4e3a\u5b83\u5b9e\u9645\u4e0a\u662f\u4e00\u4e2a\u540c\u4e49\u53cd\u590d\u3002\u6211\u4eec\u5df2\u7ecf\u77e5\u9053\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}c = c := (\\forall\\epsilon\\gt 0) (\\exists \\delta \\gt 0)(0\\lt|x-x_0|\\lt \\delta \\rightarrow |c-c|\\lt \\epsilon)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u4f46\u5b9e\u9645\u4e0a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=|c-c|\\lt \\epsilon<\/span><\/span> \u5bf9\u4e8e\u4efb\u4f55\u6b63\u7684 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span> \u90fd\u662f\u4e00\u4e2a\u540c\u4e49\u53cd\u590d\uff0c\u56e0\u6b64\u6574\u4e2a\u8868\u8fbe\u5f0f\u4e5f\u662f\u540c\u4e49\u53cd\u590d\uff0c\u7ed3\u8bba\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}c = c<\/span><\/span>\u3002<\/p>\n<h4>\u51fd\u6570\u5546\u7684\u6781\u9650<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3563s\" rel=\"noopener\" target=\"_blank\"><strong>\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\u51fd\u6570\u5546\u7684\u6781\u9650\u89c4\u5219\u4e86\u3002<\/strong><\/a> \u5b83\u662f\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}\\frac{f(x)}{g(x)}= \\frac{L}{M}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u4e0e\u4e4b\u524d\u7684\u6027\u8d28\u7c7b\u4f3c\uff0c\u6211\u4eec\u5047\u8bbe\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}=\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}g(x) = M\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u8bc1\u660e<\/strong><\/p>\n<p style=\"text-align: justify;\">\u5e78\u8fd0\u7684\u662f\uff0c\u6211\u4eec\u4e0d\u9700\u8981\u518d\u50cf\u4e4b\u524d\u4e00\u6837\u505a\u7e41\u7410\u7684\u8bc1\u660e\uff0c\u56e0\u4e3a\u6211\u4eec\u73b0\u5728\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u8fd9\u4e9b\u7ed3\u679c\u3002\u4f46\u5728\u6b64\u4e4b\u524d\uff0c\u9996\u5148\u8bc1\u660e\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}\\frac{1}{g(x)} = \\frac{1}{M}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u4e3a\u4e86\u8bc1\u660e\u8fd9\u4e00\u70b9\uff0c\u6211\u4eec\u53ea\u9700\u4f7f\u7528\u51fd\u6570\u4e58\u79ef\u7684\u6781\u9650\u89c4\u5219\u548c\u5e38\u6570\u51fd\u6570\u7684\u6781\u9650\u89c4\u5219\uff0c\u5e76\u786e\u4fdd <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x)<\/span><\/span> \u4e0d\u4e3a\u96f6\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 1 = \\lim_{x\\to x_0}\\left( 1 \\right) \\lim_{x\\to x_0}\\left( g(x) \\cdot \\frac{1}{g(x)} \\right) = \\lim_{x\\to x_0}g(x) \\cdot \\lim_{x\\to x_0} \\frac{1}{g(x)} = M \\cdot \\lim_{x\\to x_0} \\frac{1}{g(x)}<\/span><\/span><\/p>\n<p style=\"text-align: center;\">\u56e0\u6b64\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\frac{1}{g(x)} = \\frac{1}{M}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u6700\u7ec8\uff0c\u6839\u636e\u51fd\u6570\u4e58\u79ef\u7684\u6781\u9650\u89c4\u5219\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\frac{f(x)}{g(x)} = \\lim_{x\\to x_0} f(x) \\frac{1}{g(x)}= L \\cdot\\frac{1}{M} = \\frac{L}{M}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u8fd9\u6210\u7acb\u7684\u524d\u63d0\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M<\/span><\/span> \u4e0d\u4e3a\u96f6\u3002<\/p>\n<h4>\u81ea\u7136\u5e42\u6b21\u7684\u6781\u9650<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3725s\" rel=\"noopener\" target=\"_blank\"><strong>\u8fd9\u4e2a\u6027\u8d28\u544a\u8bc9\u6211\u4eec\uff0c<\/strong><\/a> \u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x_0 \\to x_0}f(x) = L<\/span><\/span>\uff0c\u90a3\u4e48\u5c06\u4f1a\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall n \\in \\mathbb{N}\\right) \\left( \\lim_{x\\to x_0} \\left( [f(x)]^n \\right) = L^n \\right)<\/span><\/span>\u3002\u6211\u4eec\u53ef\u4ee5\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e\u8fd9\u4e00\u70b9\u3002<\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u8bc1\u660e\uff1a<\/strong><\/p>\n<ul style=\"text-align: justify;\">\n<li><strong>\u60c5\u51b5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=1<\/span><\/span>\uff1a<\/strong> (\u521d\u59cb\u6b65\u9aa4)\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^1 = \\lim_{x\\to x_0} f(x) = L.<\/span><\/span> \u8fd9\u5b8c\u6210\u4e86\u521d\u59cb\u6b65\u9aa4 \u2705<\/p>\n<\/li>\n<li><strong>\u60c5\u51b5 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=k<\/span><\/span>\uff1a<\/strong> (\u5f52\u7eb3\u6b65\u9aa4)\n<p style=\"text-align: justify;\">\u5047\u8bbe\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^k = L^k <\/span><\/span> (\u5f52\u7eb3\u5047\u8bbe)\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u8bc1\u660e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^{k+1} = L^{k+1} <\/span><\/span>\u3002<\/p>\n<p style=\"text-align: justify;\">\u6211\u4eec\u6709\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^{k+1} = \\lim_{x\\to x_0} \\{f(x) [f(x)]^k\\} = \\lim_{x\\to x_0}f(x) \\lim_{x\\to x_0} [f(x)]^{k} =L \\lim_{x\\to x_0} [f(x)]^{k}<\/span><\/span>\u3002\u8fd9\u662f\u57fa\u4e8e\u524d\u9762\u5df2\u7ecf\u8bc1\u660e\u7684\u4e58\u79ef\u6781\u9650\u89c4\u5219\u3002<\/p>\n<p style=\"text-align: justify;\">\u7136\u540e\uff0c\u6839\u636e\u5f52\u7eb3\u5047\u8bbe\uff0c\u6211\u4eec\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^{k+1} = L \\lim_{x\\to x_0} [f(x)]^{k} =L\\cdot L^k = L^{k+1}.<\/span><\/span> \u8fd9\u5b8c\u6210\u4e86\u5f52\u7eb3\u6b65\u9aa4 \u2705<\/p>\n<\/li>\n<li>\u56e0\u6b64\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall n \\in \\mathbb{N}\\right) \\left( \\lim_{x\\to x_0} \\left( [f(x)]^n \\right) = L^n \\right). <\/span><\/span><\/li>\n<\/ul>\n<h4>n\u6b21\u65b9\u6839\u7684\u6781\u9650<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3912s\" rel=\"noopener\" target=\"_blank\"><strong>\u7c7b\u4f3c\u4e8e\u5e42\u6b21\uff0c\u6211\u4eec\u6709\uff1a<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall n \\in \\mathbb{N}\\right) \\left( \\lim_{x\\to x_0} \\sqrt[n]{f(x)} = \\sqrt[n]{L} \\right) <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>\u8bc1\u660e\uff1a<\/strong><\/p>\n<p style=\"text-align: justify;\">\u4f7f\u7528\u521a\u521a\u8bc1\u660e\u7684\u5e42\u6b21\u89c4\u5219\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle L= \\lim_{x\\to x_0} f(x)=\\lim_{x\\to x_0} \\left[\\sqrt[n]{f(x)}\\right]^n = \\left[ \\lim_{x\\to x_0} \\sqrt[n]{f(x)}\\right]^n <\/span><\/span><\/p>\n<p style=\"text-align: center;\">\u56e0\u6b64\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\sqrt[n]{f(x)} =\\sqrt[n]{L}.<\/span><\/span><\/p>\n<h4>\u5206\u6570\u5e42\u6b21\u7684\u6781\u9650<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4007s\" rel=\"noopener\" target=\"_blank\"><strong>\u7ed3\u5408\u4e4b\u524d\u4e24\u4e2a\u8bc1\u660e\u7684\u529b\u91cf<\/strong><\/a> \u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u6700\u540e\u4e00\u4e2a\u8bc1\u660e\uff0c\u5373\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall p,q\\neq 0 \\in \\mathbb{Z}\\right) \\left( \\lim_{x\\to x_0} \\left[f(x)\\right]^{\\frac{p}{q}} = L^{\\frac{p}{q}} \\right). <\/span><\/span> \u8fd9\u4e2a\u7ed3\u679c\u662f\u901a\u8fc7\u4e58\u79ef\u89c4\u5219\u5f97\u5230\u7684\uff0c\u56e0\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle [f(x)]^{\\frac{p}{q}} =[\\sqrt[q]{f(x)}]^p <\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle L^{\\frac{p}{q}} =[\\sqrt[q]{L}]^p. <\/span><\/span><\/p>\n<h4>\u6781\u9650 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}x = x_0<\/span><\/span><\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4073s\" rel=\"noopener\" target=\"_blank\"><strong>\u8fd9\u4e2a\u8bc1\u660e\u5c06\u7ed3\u675f\u8fd9\u4e00\u7cfb\u5217\u7684\u8bc1\u660e\u3002<\/strong><\/a> \u7ed3\u5408\u4e4b\u524d\u7684\u8bc1\u660e\uff0c\u6211\u4eec\u5c06\u80fd\u591f\u76f4\u89c2\u5730\u8ba1\u7b97\u5927\u91cf\u7684\u6781\u9650\u3002<\/p>\n<p style=\"text-align: justify;\">\u8bc1\u660e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}x = x_0<\/span><\/span> \u975e\u5e38\u7b80\u5355\uff0c\u56e0\u4e3a\u8981\u6ee1\u8db3\u8fd9\u4e2a\u6761\u4ef6\uff0c\u9700\u8981\uff1a<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0) (\\exists \\delta \\gt 0)(0\\lt |x-x_0|\\lt \\delta\\rightarrow |x-x_0|\\lt \\epsilon)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">\u6839\u636e\u6781\u9650\u5b9a\u4e49\uff0c\u5bf9\u4e8e\u4efb\u610f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span>\uff0c\u5fc5\u987b\u81f3\u5c11\u5b58\u5728\u4e00\u4e2a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u4f7f\u5f97\u5176\u4ed6\u6761\u4ef6\u6210\u7acb\uff1b\u56e0\u6b64\uff0c\u53ea\u9700\u627e\u5230\u8fd9\u6837\u4e00\u4e2a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> \u6765\u9a8c\u8bc1\u6781\u9650\u7684\u6b63\u786e\u6027\u3002\u8fd9\u5b9e\u9645\u4e0a\u662f\u663e\u800c\u6613\u89c1\u7684\uff0c\u56e0\u4e3a\u4efb\u4f55 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta\\leq\\epsilon<\/span><\/span> \u90fd\u6ee1\u8db3\u6b64\u6761\u4ef6\u3002\u56e0\u6b64\uff1a<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}x = x_0.<\/span><\/span><\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h2>\u7b80\u5355\u6781\u9650\u7684\u8ba1\u7b97<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4155s\" rel=\"noopener\" target=\"_blank\"><strong>\u611f\u8c22\u6211\u4eec\u521a\u521a\u56de\u987e\u7684\u6240\u6709\u8fd9\u4e9b\u5b9a\u7406<\/strong><\/a>\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u76f8\u5bf9\u76f4\u89c2\u7684\u65b9\u5f0f\u8ba1\u7b97\u5927\u91cf\u6781\u9650\uff0c\u5c31\u50cf\u6211\u4eec\u53ea\u9700\u5bf9\u51fd\u6570\u8fdb\u884c\u6c42\u503c\u4e00\u6837\u3002\u8fd9\u91cc\u4f60\u53ef\u4ee5\u770b\u5230\u4e00\u4e9b\u4f8b\u5b50\uff1a<\/p>\n<ol style=\"text-align:left;\">\n<li>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{}\\\\ \\begin{array}{rl}\n\n \\displaystyle \\lim_{x\\to 2}(x^2 + 4x) &amp; = \\displaystyle \\lim_{x\\to 2}(x^2) + \\lim_{x\\to 2}(4x) \\\\ \\\\\n\n&amp; = \\displaystyle \\left(\\lim_{x\\to 2} x \\right)^2 + 4\\lim_{x\\to 2} x \\\\ \\\\\n\n&amp; = (2)^2 + 8 = 12\n\n\\end{array}<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{} \\\\ \\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to 1}\\left.\\frac{(3x-1)^2}{(x+1)^3} \\right. &amp; = \\displaystyle \\frac{(3(1)-1)^2}{((1)+1)^3} \\\\ \\\\\n\n&amp; = \\displaystyle \\frac{4}{8} = \\frac{1}{2}\n\n\\end{array}\n\n<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{} \\\\ \\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to 2} \\frac{x-2}{x^2 - 4} &amp;= \\displaystyle \\lim_{x\\to 2} \\frac{x-2}{(x-2)(x+2)} \\\\ \\\\\n\n&amp; = \\displaystyle \\lim_{x\\to 2} \\frac{1}{x+2} = \\dfrac{1}{4}\n\n\\end{array}\n\n <\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{} \\\\ \\begin{array}{rl}\n\n\\displaystyle \\lim_{h\\to 0} \\frac{(x+h)^3-x^3}{h} &amp;= \\displaystyle \\lim_{h\\to 0} \\frac{x^3 + 3x^2 h + 3xh^2 -x^3}{h} \\\\ \\\\\n\n&amp; = \\displaystyle\\lim_{h\\to 0} \\frac{3x^3 h + 3xh^2}{h} \\\\ \\\\\n\n&amp; = \\displaystyle \\lim_{h\\to 0} 3x^2 + 3xh = 3x^2\n\n\\end{array}\n\n <\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{} \\\\ \\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to 1} \\frac{x-1}{\\sqrt{x^2 + 3} - 2 } &amp;=\\displaystyle \\lim_{x\\to 1} \\frac{x-1}{\\sqrt{x^2 + 3} - 2 } \\frac{\\sqrt{x^2 + 3} + 2}{\\sqrt{x^2 + 3} + 2} \\\\ \\\\\n\n&amp; =\\displaystyle \\lim_{x\\to 1} \\frac{(x-1)(\\sqrt{x^2 + 3} + 2)}{(x^2 + 3) - 4 } \\\\ \\\\\n\n&amp; =\\displaystyle \\lim_{x\\to 1} \\frac{(x-1)(\\sqrt{x^2 + 3} + 2)}{x^2 -1 } \\\\ \\\\\n\n&amp; =\\displaystyle \\lim_{x\\to 1} \\frac{(x-1)(\\sqrt{x^2 + 3} + 2)}{(x-1)(x+1) } \\\\ \\\\\n\n&amp; =\\displaystyle \\lim_{x\\to 1} \\frac{\\sqrt{x^2 + 3} + 2}{ x+1 } \\\\ \\\\\n\n&amp; =\\displaystyle \\frac{2+2}{2} =2\n\n\\end{array}<\/span><\/span><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u4e00\u5143\u5b9e\u53d8\u91cf\u51fd\u6570\u7684\u6781\u9650 \u6458\u8981\uff1a \u672c\u8282\u8bfe\u6df1\u5165\u8ba8\u8bba\u4e86\u4e00\u5143\u5b9e\u53d8\u91cf\u51fd\u6570\u7684\u6781\u9650\u7684\u6b63\u5f0f\u5b9a\u4e49\uff0c\u5e76\u5728\u6b64\u57fa\u7840\u4e0a\u8bc1\u660e\u4e86\u6781\u9650\u4ee3\u6570\u7684\u4e3b\u8981\u6027\u8d28\u3002 \u5b66\u4e60\u76ee\u6807\uff1a \u672c\u8282\u8bfe\u7ed3\u675f\u65f6\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a \u8bb0\u4f4f\u4e00\u5143\u5b9e\u53d8\u91cf\u51fd\u6570\u7684\u6781\u9650\u5b9a\u4e49\u3002 \u901a\u8fc7\u63a8\u5bfc\u8bc1\u660e\u5bfc\u81f4\u6781\u9650\u4ee3\u6570\u7684\u6027\u8d28\u3002 \u4f7f\u7528\u6781\u9650\u4ee3\u6570\u53ca\u5176\u6027\u8d28\u8ba1\u7b97\u4e00\u5143\u5b9e\u53d8\u91cf\u51fd\u6570\u7684\u6781\u9650\u3002 \u76ee\u5f55 \u4ecb\u7ecd \u4ece\u56fe\u5f62\u89d2\u5ea6\u770b\u51fd\u6570\u6781\u9650\u7684\u76f4\u89c2\u6982\u5ff5 \u6781\u9650\u7684\u6b63\u5f0f\u5b9a\u4e49 \u6781\u9650\u7684\u6027\u8d28 \u5982\u679c\u6781\u9650\u5b58\u5728\uff0c\u90a3\u4e48\u5b83\u662f\u552f\u4e00\u7684 \u6781\u9650\u4ee3\u6570 \u7b80\u5355\u6781\u9650\u8ba1\u7b97 \u4ecb\u7ecd \u4ee3\u6570\u548c\u51e0\u4f55\u4e0e\u5fae\u79ef\u5206\u7684\u5b66\u4e60\u6709\u4ec0\u4e48\u4e0d\u540c\uff1f \u8fd9\u4e2a\u95ee\u9898\u7684\u7b54\u6848\u7531\u6781\u9650\u7684\u6982\u5ff5\u7ed9\u51fa\u3002\u56e0\u6b64\uff0c\u672c\u6587\u7ae0\u8ba8\u8bba\u4e86\u6781\u9650\u53ca\u5176\u5b9a\u4e49\u3002 \u6211\u4eec\u901a\u5e38\u5c06\u201c\u6781\u9650\u201d\u4e00\u8bcd\u4e0e\u67d0\u79cd\u8fb9\u754c\u8054\u7cfb\u8d77\u6765\uff0c\u5c31\u50cf\u6709\u7aef\u70b9a\u548cb\u7684\u533a\u95f4\u7684\u8fb9\u754c\uff08\u65e0\u8bba\u5176\u6027\u8d28\u5982\u4f55\uff09\u3002 , \u6216\u8005\u50cf\u201c\u73b0\u5728\u201d\uff0c\u6211\u4eec\u53ef\u4ee5\u8bf4\u5b83\u662f\u8fc7\u53bb\u548c\u672a\u6765\u7684\u8fb9\u754c\u3002\u7c7b\u4f3c\u5730\uff0c\u6781\u9650\u7684\u6982\u5ff5\u5f15\u5165\u4e86\u5bf9\u8fd9\u4e00\u76f4\u89c2\u60f3\u6cd5\u7684\u6570\u5b66\u7406\u89e3\uff0c\u5373\u6e10\u8fd1\u5730\u63a5\u8fd1\u67d0\u4e2a\u70b9\u3002 \u4ece\u56fe\u5f62\u89d2\u5ea6\u770b\u51fd\u6570\u6781\u9650\u7684\u76f4\u89c2\u6982\u5ff5 \u4e3a\u4e86\u5f00\u59cb\u76f4\u89c2\u5730\u7406\u89e3\u6781\u9650\u7684\u6982\u5ff5\uff0c\u6700\u597d\u4ece\u51fd\u6570\u7684\u56fe\u5f62\u8868\u793a\u5165\u624b\uff0c\u5e76\u8be2\u95ee\u5f53 \u968f \u63a5\u8fd1 \u65f6\u5c06\u4f1a\u53d1\u751f\u4ec0\u4e48\u3002 \u5982\u679c \u63a5\u8fd1 \uff0c\u90a3\u4e48\u5c06\u5b58\u5728\u4ee5 \u4e3a\u4e2d\u5fc3\uff0c\u534a\u5f84\u4e3a \u7684\u5f00\u533a\u95f4\uff0c\u4f7f\u5f97 \u5305\u542b\u5728\u5176\u4e2d\u3002\u6211\u4eec\u53ef\u4ee5\u7528\u4e09\u79cd\u4e0d\u540c\u7684\u65b9\u5f0f\u6765\u8868\u793a\uff1a , , \u6216\u8005 \u5728\u6211\u4eec\u7684\u4e0a\u4e0b\u6587\u4e2d\uff0c\u8fd9\u4e09\u79cd\u8868\u793a\u65b9\u5f0f\u90fd\u662f\u76f8\u540c\u7684\uff1b\u4f46\u6700\u540e\u4e00\u79cd\uff0c\u5373\u201c \u5305\u542b\u5728\u4ee5 \u4e3a\u4e2d\u5fc3\uff0c\u534a\u5f84\u4e3a \u7684\u5f00\u7403\u4e2d\u201d\uff0c\u66f4\u9002\u5408\u7528\u4e8e \u62d3\u6251\u5b66\u8bfe\u7a0b\uff0c\u5728\u90a3\u91cc\u4f1a\u5bf9\u8fd9\u4e2a\u201c\u90bb\u8fd1\u95ee\u9898\u201d\u8fdb\u884c\u66f4\u6df1\u5165\u7684\u63a2\u8ba8\u3002 \u5982\u679c\u53d1\u751f\u8fd9\u79cd\u60c5\u51b5\uff0c\u90a3\u4e48\u6211\u4eec\u5c06\u770b\u5230\u5b58\u5728\u53e6\u4e00\u4e2a\u4ee5 \u4e3a\u4e2d\u5fc3\uff0c\u534a\u5f84\u4e3a \u7684\u5f00\u533a\u95f4\uff0c\u4f7f\u5f97 \u5305\u542b\u5728\u5176\u4e2d\uff0c\u5373\uff1a\u3002 \u8fd9\u91cc\u4ea7\u751f\u4e86\u6781\u9650\u6982\u5ff5\u7684\u57fa\u672c\u601d\u60f3\uff0c\u5373\u5f53 \u65f6\uff0c\u5982\u679c \uff0c\u5219\u6781\u9650 \u5b58\u5728\u3002 \u6781\u9650\u7684\u6b63\u5f0f\u5b9a\u4e49 \u4ece\u521a\u624d\u5448\u73b0\u7684\u76f4\u89c2\u548c\u56fe\u5f62\u6982\u5ff5\u5f00\u59cb\uff0c\u53ef\u4ee5\u63a8\u5bfc\u51fa\u6781\u9650\u7684\u6b63\u5f0f\u5b9a\u4e49\u3002\u6211\u4eec\u8bf4\u6781\u9650\u5b58\u5728\uff0c\u5f53\u4e14\u4ec5\u5f53\u65e0\u8bba \u5982\u4f55\uff08\u5373 \u548c \u4e4b\u95f4\u7684\u8ddd\u79bb\uff09\uff0c\u603b\u80fd\u627e\u5230\u4e00\u4e2a \uff0c\u4f7f\u5f97\u5f53 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27786,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":5,"footnotes":""},"categories":[858,575],"tags":[],"class_list":["post-27819","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-858","category-575"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u5b9e\u53d8\u91cf\u51fd\u6570\u7684\u6781\u9650 - 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