{"id":19,"date":"2023-09-17T11:08:30","date_gmt":"2023-09-17T11:08:30","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e4%b8%89%e8%a7%92%e6%9e%81%e9%99%90\/"},"modified":"2023-09-17T11:08:30","modified_gmt":"2023-09-17T11:08:30","slug":"%e4%b8%89%e8%a7%92%e6%9e%81%e9%99%90","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e4%b8%89%e8%a7%92%e6%9e%81%e9%99%90\/","title":{"rendered":"\u4e09\u89d2\u6781\u9650"},"content":{"rendered":"<p>\u5728\u8fd9\u91cc\u60a8\u5c06\u4e86\u89e3\u5982\u4f55\u6c42\u89e3\u4e09\u89d2\u6781\u9650\u3002\u60a8\u5c06\u80fd\u591f\u770b\u5230\u4e09\u89d2\u51fd\u6570\u6781\u9650\u7684\u51e0\u4e2a\u793a\u4f8b\uff0c\u751a\u81f3\u53ef\u4ee5\u901a\u8fc7\u89e3\u51b3\u4e09\u89d2\u51fd\u6570\u6781\u9650\u7684\u5206\u6b65\u7ec3\u4e60\u8fdb\u884c\u7ec3\u4e60\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-los-limites-trigonometricos\"><\/span>\u4ec0\u4e48\u662f\u4e09\u89d2\u6781\u9650\uff1f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>\u4e09\u89d2\u6781\u9650\u662f\u6839\u636e\u4e09\u89d2\u51fd\u6570\u8ba1\u7b97\u7684\u6781\u9650\u3002<\/strong>\u4e3a\u4e86\u89e3\u51b3\u4e09\u89d2\u6781\u9650\uff0c\u5fc5\u987b\u5e94\u7528\u521d\u6b65\u7a0b\u5e8f\uff0c\u56e0\u4e3a\u5b83\u4eec\u901a\u5e38\u4f1a\u5f15\u8d77\u4e0d\u786e\u5b9a\u6027\u3002<\/p>\n<p>\u6b64\u5916\uff0c\u4e09\u89d2\u51fd\u6570\u7684\u65e0\u9650\u6781\u9650\u4e0d\u5b58\u5728\uff0c\u56e0\u4e3a\u5b83\u4eec\u662f\u5468\u671f\u51fd\u6570\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5b83\u7684\u56fe\u8868\u4e0d\u65ad\u5468\u671f\u6027\u5730\u91cd\u590d\uff0c\u800c\u4e0d\u8d8b\u5411\u4e8e\u7279\u5b9a\u503c\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formulas-de-los-limites-trigonometricos\"><\/span>\u4e09\u89d2\u6781\u9650\u516c\u5f0f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6240\u6709\u4e09\u89d2\u6781\u9650\u5747\u901a\u8fc7\u4ee5\u4e0b\u4e24\u4e2a\u516c\u5f0f\u8ba1\u7b97\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-14300dcf4010d732b8568b9b4460b5e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"116\" 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>\u914d\u65b9\u6f14\u793a<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5982\u679c\u6211\u4eec\u5c1d\u8bd5\u901a\u8fc7\u66ff\u6362\u6765\u8ba1\u7b97\u6781\u9650\uff0c\u6211\u4eec\u4f1a\u5f97\u5230\u96f6\u4e4b\u95f4\u7684\u96f6\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-0ba13ab3640b429e546e97da2a0ab155_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}=\\frac{\\text{sen}(0)}{0}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"193\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4f46\u8fd9\u4e2a\u4e09\u89d2\u516c\u5f0f\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u66f4\u63a5\u8fd1\u7684\u51fd\u6570\u503c\u548c\u66f4\u63a5\u8fd1x=0\uff08\u4ee5\u5f27\u5ea6\u8868\u793a\u7684\u89d2\u5ea6\uff09\u6765\u8bc1\u660e\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-2a7ed3df9110a97c224bde10980f2682_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{\\text{sen}(x)}{x}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"142\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-123\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ccd668acef73b9140a0cbbb9c1d53ad3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\begin{array}{l}f(-1)=\\cfrac{\\text{sen}(-1)}{-1}=0,84147\\\\[3ex]f(-0,1)=\\cfrac{\\text{sen}(-0,1)}{-0,1}=0,99833\\\\[3ex]f(-0,01)=\\cfrac{\\text{sen}(-0,01)}{-0,01}=0,99998\\\\[3ex]f(-0,001)=\\cfrac{\\text{sen}(-0,001)}{-0,001}=0,99999\\end{array}\\\\[14ex]\\vdots\\\\[2ex]\\displaystyle\\lim_{x\\to 0^-}\\frac{\\text{sen}(x)}{x}=1\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"312\" width=\"288\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-66152efc3ce1fa761186a65db677af27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\begin{array}{l}f(1)=\\cfrac{\\text{sen}(1)}{1}=0,84147\\\\[3ex]f(0,1)=\\cfrac{\\text{sen}(0,1)}{0,1}=0,99833\\\\[3ex]f(0,01)=\\cfrac{\\text{sen}(0,01)}{0,01}=0,99998\\\\[3ex]f(0,001)=\\cfrac{\\text{sen}(0,001)}{0,001}=0,99999\\end{array}\\\\[14ex]\\vdots\\\\[2ex]\\displaystyle\\lim_{x\\to 0^+}\\frac{\\text{sen}(x)}{x}=1\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"312\" width=\"261\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u4e09\u89d2\u51fd\u6570\u7684\u4e24\u4e2a\u6a2a\u5411\u6781\u9650\u4e3a 1\uff0c\u56e0\u6b64 x=0 \u70b9\u5904\u7684\u6781\u9650\u4e3a 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-8af649189957b154866097e315f7cb92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\displaystyle\\lim_{x\\to 0^-}\\frac{\\text{sen}(x)}{x}=\\lim_{x\\to 0^+}\\frac{\\text{sen}(x)}{x}=1\\\\[3ex]\\color{orange}\\bm{\\downarrow}\\\\[2ex]\\lim_{x\\to 0}\\displaystyle\\frac{\\text{sen}(x)}{x}=1\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"130\" width=\"243\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u5f53 x \u8d8b\u4e8e 0 \u65f6\uff0cx \u7684\u6b63\u5f26\u9664\u4ee5 x \u7684\u4e09\u89d2\u6781\u9650\u7b49\u4e8e 1\u3002<\/p>\n<p class=\"has-text-align-left\">\u8fd9\u4e2a\u516c\u5f0f\u4e5f\u53ef\u4ee5\u5e94\u7528\u4e8e\u591a\u4e2a\u89d2\u5ea6\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-e8b276a8e0f8bf93f3ea2b7d0158adbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(kx)}{kx}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"125\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-51835176b411ca1ec4f37835a83685fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{x}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"146\" 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>\u914d\u65b9\u6f14\u793a<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5982\u679c\u6211\u4eec\u5c1d\u8bd5\u901a\u8fc7\u76f4\u63a5\u66ff\u6362\u6765\u6c42\u6781\u9650\uff0c\u6211\u4eec\u4f1a\u5f97\u5230\u96f6\u4e4b\u95f4\u7684\u4e0d\u5b9a\u5f62\u5f0f\u96f6\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-3e50a5f25f1ca148a4e0107e75e62c43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{x}=}\\frac{1-\\text{cos}(0)}{0}=\\frac{1-1}{0}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"319\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4f46\u6211\u4eec\u53ef\u4ee5\u4ece\u4e0a\u9762\u7684\u516c\u5f0f\u6765\u68c0\u9a8c\u662f\u5426\u76f8\u7b49\u3002\u4e3a\u6b64\uff0c\u60a8\u5fc5\u987b\u9996\u5148\u5c06\u5206\u6570\u7684\u5206\u5b50\u548c\u5206\u6bcd\u4e58\u4ee5 1 \u52a0\u4e0a x \u7684\u4f59\u5f26\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-40196b4f425393970ff11577ef645dba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\bigl(1-\\text{cos}(x)\\bigr)\\cdot \\bigl(1+\\text{cos}(x)\\bigr)}{x\\cdot \\bigl(1+\\text{cos}(x)\\bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"235\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u6211\u4eec\u5728\u5206\u6570\u7684\u5206\u5b50\u4e2d\u6709\u663e\u7740\u7684\u6052\u7b49\u5f0f\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u7b80\u5316\u5b83\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-052fdb01d818c3baa3293d4e1927d37c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1^2-\\text{cos}^2(x)}{x\\cdot \\bigl(1+\\text{cos}(x)\\bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"151\" style=\"vertical-align: -20px;\"><\/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-4b7f2f3a29d5eaebb5f226607e80dbb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}^2(x)}{x\\cdot \\bigl(1+\\text{cos}(x)\\bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"151\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4ece\u57fa\u672c\u7684\u4e09\u89d2\u6052\u7b49\u5f0f\u5f00\u59cb\uff0c\u6211\u4eec\u91cd\u5199\u5206\u5b50\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-ff23bded6a6a479ee358e635c74ef2fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}^2(x)+\\text{cos}^2(x)=1 \\ \\longrightarrow \\ \\text{sen}^2(x)=1-\\text{cos}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"381\" style=\"vertical-align: -5px;\"><\/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-df11237bdbf1c1ef5f607f08db54ca91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}^2(x)}{x\\cdot \\bigl(1+\\text{cos}(x)\\bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"151\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5206\u6570\u8f6c\u6362\u4e3a\u5206\u6570\u7684\u4e58\u79ef\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-cff26553bbe1117d69b5a11e0371b996_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)\\cdot \\text{sen}(x)}{x\\cdot \\bigl(1+\\text{cos}(x)\\bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"151\" style=\"vertical-align: -20px;\"><\/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-ac46678f2030eed0dc15696613ec60ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot \\frac{\\text{sen}(x)}{1+\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"178\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5229\u7528\u6781\u9650\u7684\u6027\u8d28\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e0a\u9762\u7684\u8868\u8fbe\u5f0f\u8f6c\u6362\u4e3a\u6781\u9650\u7684\u4e58\u79ef\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-94b986a7b575a61eeba306ce22a6a01e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot \\lim_{x\\to 0}\\frac{\\text{sen}(x)}{1+\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"209\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4f7f\u7528\u4e0a\u9762\u6f14\u793a\u7684\u516c\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u8f7b\u677e\u7b80\u5316\u4e09\u89d2\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-14300dcf4010d732b8568b9b4460b5e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"116\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c46c6322634327f17aa601618460fd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 1\\cdot \\lim_{x\\to 0}\\frac{\\text{sen}(x)}{1+\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"133\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b446be13fe115291c38a7c34c192d571_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 0}\\frac{\\text{sen}(x)}{1+\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"112\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u8ba1\u7b97\u7ed3\u679c\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bee04c527b0609fc39a7729ec6677874_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 0}\\frac{\\text{sen}(0)}{1+\\text{cos}(0)}=\\frac{0}{1+1}=\\frac{0}{2}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"248\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u4e09\u89d2\u6781\u9650\u516c\u5f0f\u5f97\u5230\u9a8c\u8bc1\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-51835176b411ca1ec4f37835a83685fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{x}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"146\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e0e\u5176\u4ed6\u516c\u5f0f\u4e00\u6837\uff0c\u5b83\u4e5f\u53ef\u4ee5\u7528\u4e8e\u591a\u4e2a\u89d2\u5ea6\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-afd4adcffaaad5d5b5c7063ec3542b5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}(kx)}{kx}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"156\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p>\u56e0\u6b64\uff0c<strong>\u8981\u6c42\u89e3\u4e09\u89d2\u6781\u9650\uff0c\u5c31\u5fc5\u987b\u7528\u7b97\u672f\u5bf9\u51fd\u6570\u8fdb\u884c\u53d8\u6362\uff0c\u5f97\u5230\u7c7b\u4f3c\u7684\u8868\u8fbe\u5f0f\u3002<\/strong>\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u4f7f\u7528\u4e24\u4e2a\u516c\u5f0f\u4e4b\u4e00\u5e76\u627e\u5230\u6781\u9650\u503c\u3002<\/p>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u6709\u65f6\u6211\u4eec\u53ef\u80fd\u9700\u8981\u5e94\u7528\u67d0\u4e9b\u4e09\u89d2\u6052\u7b49\u5f0f\uff0c\u56e0\u6b64\u6211\u4eec\u5c06\u4ee5\u4e0b\u6240\u6709\u516c\u5f0f\u7559\u7ed9\u60a8<\/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>\u4e09\u89d2\u6052\u7b49\u5f0f<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u8fde\u63a5\u4e09\u4e2a\u4e3b\u8981\u4e09\u89d2\u6bd4\u7387\u7684\u516c\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dbf6d65fa67f0a2161bd99ee7431f015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(x)=\\cfrac{\\text{sen}(x)}{\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"124\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u57fa\u672c\u4e09\u89d2\u6052\u7b49\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-92d80771f891319379b2e756c5524aaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}^2(x)+\\text{cos}^2(x)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"165\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u7531\u57fa\u672c\u4e09\u89d2\u5173\u7cfb\u5bfc\u51fa\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-983ec3f9bdead575a110ab13a3149351_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1+\\text{tan}^2 (x)=\\cfrac{1}{\\text{cos}^2(x)}=\\text{sec}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"245\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7b04254cbdd6156ce5fd5449f5234a9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1+\\text{cot}^2 (x)=\\cfrac{1}{\\text{sen}2(x)}=\\text{cosec}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"262\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5bf9\u89d2\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-a0a5345d1ad85390cacfc38e99beb548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}(-x)=-\\text{sen}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -5px;\"><\/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-b7ef3e0d227838cf04c0f7413d1e07f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cos}(-x)=\\text{cos}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -5px;\"><\/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-507f1aa7df63922130ea766d03aaf91a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(-x)=-\\text{tan}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e24\u4e2a\u89d2\u5ea6\u4e4b\u548c\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-58668a0aaa63a0e4c39b859619d2444a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}(x+y)=\\text{sen}(x)\\text{cos}(y)+\\text{cos}(x)\\text{sen}(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"315\" style=\"vertical-align: -5px;\"><\/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-46aaa0aea1219b24ef354afcc8a15953_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cos}(x+y)=\\text{cos}(x)\\text{cos}(y)-\\text{sen}(x)\\text{sen}(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"314\" style=\"vertical-align: -5px;\"><\/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-c5c001a17b792285beacf6cf91f93033_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(x+y)=\\cfrac{\\text{tan}(x)+\\text{tan}(y)}{1-\\text{tan}(x)\\text{tan}(y)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"235\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e24\u4e2a\u89d2\u5ea6\u4e4b\u5dee\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02420c87f520da509e0193dab4798f55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}(x-y) = \\text{sen}(x)\\text{cos}(y)-\\text{cos}(x)\\text{sen}(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"315\" style=\"vertical-align: -5px;\"><\/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-9d5d84d5fa7db15e90131596953bedb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cos}(x-y) = \\text{cos}(x)\\text{cos}(y)+ \\text{sen}(x) sen(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"317\" style=\"vertical-align: -5px;\"><\/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-b5d01deaacd8e294bcfd6b6284231fa2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(x-y)=\\cfrac{\\text{tan}(x)-\\text{tan}(y)}{1+\\text{tan}(x)\\text{tan}(y)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"235\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u53cc\u89d2\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-c135a8fb824883a8b8f9ff27a737a9d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}(2x) = 2\\text{sen}(x)\\text{cos}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"185\" style=\"vertical-align: -5px;\"><\/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-91843029bf168eab0615f3bb849f2dd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cos}(2x) =\\text{cos}^2(x)-\\text{sen}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"213\" style=\"vertical-align: -5px;\"><\/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-0cc8aed863858c3052e1dae8bdcdb377_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(2x) =\\cfrac{2\\text{tan}(x)}{1-\\text{tan}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"172\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u534a\u89d2\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-ccae2dc8b2bc812d68f9361538ebaf4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{sen}\\left(\\frac{x}{2}\\right) = \\pm \\sqrt{\\frac{1-\\text{cos}(x)}{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"201\" style=\"vertical-align: -13px;\"><\/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-6f87b3d54e5b0d7527bf38b2a7a71928_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{cos}\\left(\\frac{x}{2}\\right) = \\pm \\sqrt{\\frac{1+\\text{cos}(x)}{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"200\" style=\"vertical-align: -13px;\"><\/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-1126d8d443e0285d4ecc510a119b393d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{tan}\\left(\\frac{x}{2}\\right) = \\pm \\sqrt{\\frac{1-\\text{cos}(x)}{1+\\text{cos}(x)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"202\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6b63\u5f26\u548c\u4f59\u5f26\u7684\u52a0\u6cd5\u548c\u51cf\u6cd5\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-457b2949084f43244b619fd965e403f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{sen}(x)+\\text{sen}(y)=2\\text{sen}\\left(\\frac{x+y}{2} \\right)\\text{cos}\\left(\\frac{x-y}{2} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"347\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4ff2373c8559ef4bebc00a31c7c8f2ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{sen}(x)-\\text{sen}(y)=2\\text{cos}\\left(\\frac{x+y}{2} \\right)\\text{sen}\\left(\\frac{x-y}{2} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"347\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-625d75c4be2e5bbaca73e2a3f1e1980b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{cos}(x)+\\text{cos}(y)=2\\text{cos}\\left(\\frac{x+y}{2} \\right)\\text{cos}\\left(\\frac{x-y}{2} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"344\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b0a498d3701c9c87123c269c81d266d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{cos}(x)-\\text{cos}(y)=-2\\text{sen}\\left(\\frac{x+y}{2} \\right)\\text{sen}\\left(\\frac{x-y}{2} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"360\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6b63\u5f26\u548c\u4f59\u5f26\u7684\u4e58\u79ef\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-78ab2bb9d2bc291a1f7e4c9e329d893e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{sen}(x)\\cdot \\text{sen}(y)=\\frac{1}{2}\\Bigl[\\text{cos}(x-y)-\\text{cos}(x+y)\\Bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"338\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-40daecfe989acfa36adb6772d193d027_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{cos}(x)\\cdot \\text{cos}(y)=\\frac{1}{2}\\Bigl[\\text{cos}(x+y)+\\text{cos}(x-y)\\Bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"336\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b01d45ae9c7b57d25e2bd1bdfea9dba9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{sen}(x)\\cdot \\text{cos}(y)=\\frac{1}{2}\\Bigl[\\text{sen}(x+y)+\\text{sen}(x-y)\\Bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"339\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p>\u4e3a\u4e86\u8ba9\u60a8\u80fd\u591f\u51c6\u786e\u5730\u4e86\u89e3\u4e09\u89d2\u6781\u9650\u662f\u5982\u4f55\u8ba1\u7b97\u7684\uff0c\u6211\u4eec\u5728\u4e0b\u9762\u6574\u7406\u4e86\u4e00\u4e2a\u5206\u6b65\u793a\u4f8b\u3002<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-limite-trigonometrico\"><\/span>\u4e09\u89d2\u6781\u9650\u793a\u4f8b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u8ba9\u6211\u4eec\u770b\u770b\u5982\u4f55\u4f7f\u7528\u4ee5\u4e0b\u793a\u4f8b\u6c42\u89e3\u4e09\u89d2\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-83fe05cfbae51406227f863405374405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{tan}(x)}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"83\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u5c1d\u8bd5\u8ba1\u7b97\u4e09\u89d2\u6781\u9650\uff0c\u6211\u4eec\u5f97\u5230\u96f6\u4e0e\u96f6\u4e4b\u95f4\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-29fd7943ebb7d7c4ecc7886207c4a1cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{tan}(x)}{x}=\\frac{\\text{tan}(0)}{0}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"195\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span><strong>\u8bf7\u53c2\u9605\uff1a<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/cn\/\u96f6\u4e4b\u95f4-\u96f6-0-0-\u4e0d\u786e\u5b9a\/\">\u96f6\u4e4b\u95f4\u7684\u96f6\u9650\u5236<\/a><\/span><\/p>\n<p>\u56e0\u6b64\u9700\u8981\u5bf9\u4e09\u89d2\u51fd\u6570\u8fdb\u884c\u53d8\u6362\u6765\u6c42\u89e3\u6781\u9650\u3002\u6b63\u5207\u51fd\u6570\u7b49\u4e8e\u6b63\u5f26\u9664\u4ee5\u4f59\u5f26\uff0c\u56e0\u6b64\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-dbf6d65fa67f0a2161bd99ee7431f015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(x)=\\cfrac{\\text{sen}(x)}{\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"124\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-24793b3f48b399e9fd64b2eb6758f0c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{tan}(x)}{x}=\\lim_{x\\to 0}\\frac{\\displaystyle\\frac{\\text{sen}(x)}{\\text{cos}(x)}}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"195\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p>\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5e94\u7528\u5206\u6570\u7684\u6027\u8d28\u5c06\u51fd\u6570\u8868\u793a\u4e3a\u4e58\u79ef\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-fe877e223e062371ef4aa551372cfa69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\frac{\\displaystyle\\frac{a}{b}}{\\displaystyle\\frac{c}{d}}=\\frac{a\\cdot d}{b\\cdot c}\" title=\"Rendered by QuickLaTeX.com\" height=\"69\" width=\"73\" style=\"vertical-align: -30px;\"><\/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-dd10c321c3b7dc40698b318c7187a3c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 0}\\frac{\\displaystyle\\frac{\\text{sen}(x)}{\\text{cos}(x)}}{\\displaystyle\\frac{x}{1}}=\\lim_{x\\to 0}{\\frac{\\text{sen}(x)\\cdot 1}{\\text{cos}(x) \\cdot x}=\\\\[6ex]\\displaystyle =\\lim_{x\\to 0}{\\frac{\\text{sen}(x)}{x\\text{cos}(x)}=\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot \\frac{1}{\\text{cos}(x)}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"145\" width=\"282\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u5229\u7528\u6781\u9650\u7684\u6027\u8d28\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e24\u4e2a\u76f8\u4e58\u51fd\u6570\u7684\u6781\u9650\u8f6c\u6362\u4e3a\u4e24\u4e2a\u6781\u9650\u7684\u4e58\u79ef\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-38f278cf96ac97997db5ffe530037582_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot\\frac{1}{\\text{cos}(x)}=\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot\\lim_{x\\to 0}\\frac{1}{\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"350\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u6b63\u5982\u6211\u4eec\u4e0a\u9762\u6240\u5c55\u793a\u7684\uff0c\u7b2c\u4e00\u4e2a\u4e09\u89d2\u6781\u9650\u7ed9\u51fa 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-14300dcf4010d732b8568b9b4460b5e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"116\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8ac5a01bca48ac7a961a99be694dcd8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot\\lim_{x\\to 0}\\frac{1}{\\text{cos}(x)}=1\\cdot\\lim_{x\\to 0}\\frac{1}{\\text{cos}(x)}=\\lim_{x\\to 0}\\frac{1}{\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"413\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\u53ea\u9700\u8fdb\u884c\u4ee5\u4e0b\u8ba1\u7b97\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-24b950800435e32f07649e25afd6d68e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 0}\\frac{1}{\\text{cos}(x)}=\\frac{1}{\\text{cos}(0)}=\\frac{1}{1}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"225\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-limites-trigonometricos\"><\/span>\u89e3\u51b3\u4e86\u4e09\u89d2\u6781\u9650\u7684\u7ec3\u4e60<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e601<\/h3>\n<p>\u6c42\u89e3\u4ee5\u4e0b\u4e09\u89d2\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-730128e3fffaf36349ed1c2db19d8796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{2x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"91\" 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\">\u9996\u5148\uff0c\u6211\u4eec\u5c1d\u8bd5\u901a\u8fc7\u76f4\u63a5\u6c42\u503c\u6765\u8ba1\u7b97\u4e09\u89d2\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b46144b4ff38b63a3c428b5aa60ffb5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{2x}=\\frac{\\text{sen}(4\\cdot 0)}{2\\cdot 0}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"224\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4f46\u6211\u4eec\u7684\u4e0d\u786e\u5b9a\u6027\u4e3a\u96f6\u3002\u6240\u4ee5\u6211\u4eec\u9700\u8981\u5bf9\u51fd\u6570\u8fdb\u884c\u53d8\u6362\u3002<\/p>\n<p class=\"has-text-align-left\">\u9996\u5148\uff0c\u6211\u4eec\u5c06\u901a\u8fc7\u6267\u884c\u4ee5\u4e0b\u64cd\u4f5c\u5c06 x \u4fdd\u7559\u5728\u5206\u6bcd\u4e2d\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-66cfa67d319a268be5ac3c6eaf733240_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{2x}=\\lim_{x\\to 0}\\frac{1}{2}\\cdot\\frac{\\text{sen}(4x)}{x}=\\frac{1}{2}\\cdot\\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"375\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u6211\u4eec\u5c06\u5206\u6570\u4e58\u4ee5 4 \u5e76\u9664\u4ee5 4\uff0c\u5f97\u5230\u4e00\u4e2a\u8868\u8fbe\u5f0f\uff0c\u53ef\u4ee5\u5e94\u7528\u7b2c\u4e00\u4e2a\u4e09\u89d2\u6781\u9650\u516c\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-caa05571515d0dbc716d6e8cb6b0be0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\frac{1}{2}\\lim_{x\\to 0}\\frac{\\text{sen}(4x)\\cdot 4}{x\\cdot 4}=\\frac{1}{2}\\cdot 4 \\cdot \\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{4x}=2\\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{4x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"418\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u5e94\u7528\u4e00\u5f00\u59cb\u770b\u5230\u7684\u516c\u5f0f\u6765\u6c42\u89e3\u4e09\u89d2\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e8b276a8e0f8bf93f3ea2b7d0158adbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(kx)}{kx}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"125\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e21a4ad74086fda398299e2d83c9a052_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2\\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{4x}=2\\cdot 1=\\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"190\" style=\"vertical-align: -12px;\"><\/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>\u8ba1\u7b97\u4ee5\u4e0b\u4e09\u89d2\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8b6eb2bc6fa65e96cfe55e695b93b2cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)+\\text{tan}(x)}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"153\" 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\">\u9996\u5148\uff0c\u6211\u4eec\u5c1d\u8bd5\u627e\u5230\u4e09\u89d2\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2e4a361830bf074dfb37219d1288c315_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)+\\text{tan}(x)}{x}=\\frac{\\text{sen}(0)+\\text{tan}(0)}{0}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"334\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4f46\u4e0d\u5b9a\u5f62\u5f0f\u201c\u96f6\u201d\u5bf9\u5e94\u4e8e\u201c\u96f6\u201d\u5df2\u8fbe\u5230\u3002<\/p>\n<p class=\"has-text-align-left\">\u7136\u540e\uff0c\u6211\u4eec\u5c06\u6b63\u5207\u8f6c\u6362\u4e3a\u6b63\u5f26\u548c\u4f59\u5f26\u7684\u5546\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-e8dbbefdca880d5806977d6b13b473b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)+\\text{tan}(x)}{x}=\\lim_{x\\to 0}\\frac{\\displaystyle\\text{sen}(x)+\\frac{\\text{sen}(x)}{\\text{cos}(x)}}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"335\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u4e58\u4ee5\u548c\u9664\u4ee5 x \u7684\u4f59\u5f26\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-53452abaa54f3c7e46b75965500221ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\left(\\displaystyle\\text{sen}(x)+\\frac{\\text{sen}(x)}{\\text{cos}(x)}\\right)\\cdot\\text{cos}(x)}{x\\cdot\\text{cos}(x)}=\\lim_{x\\to 0}\\frac{\\text{sen}(x)\\text{cos}(x)+\\text{sen}(x)}{x\\cdot\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"67\" width=\"469\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u5728\u5206\u5b50\u4e2d\u53d6\u4e00\u4e2a\u516c\u56e0\u6570\uff0c\u7136\u540e\u5c06\u4e09\u89d2\u6781\u9650\u4e00\u5206\u4e3a\u4e8c\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-eeaa601256f134072260480b64210950_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)(\\text{cos}(x)+1)}{x\\cdot\\text{cos}(x)}=\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot\\lim_{x\\to 0}\\frac{\\text{cos}(x)+1}{\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"408\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u6c42\u51fa\u4e09\u89d2\u6781\u9650\u7684\u7ed3\u679c\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-7259ba5ebcb847c0953947ca2fb1d219_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot\\lim_{x\\to 0}\\frac{\\text{cos}(x)+1}{\\text{cos}(x)}=1\\cdot\\frac{\\text{cos}(0)+1}{\\text{cos}(0)} =\\frac{1+1}{1}=\\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"435\" style=\"vertical-align: -17px;\"><\/p>\n<\/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>\u5f53 x \u63a5\u8fd1\u96f6\u65f6\uff0c\u6c42\u89e3\u4ee5\u4e0b\u4e09\u89d2\u51fd\u6570\u7684\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f9da1c65931e93b840821e76bc20d629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{tan}(x)-\\text{sen}{(x)}}{3x\\cdot\\text{tan}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -17px;\"><\/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\">\u901a\u8fc7\u76f4\u63a5\u8ba1\u7b97\uff0c\u6211\u4eec\u5f97\u5230 0 \u4e4b\u95f4\u7684\u4e0d\u786e\u5b9a\u6781\u9650 0\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67e20b7fd699b38122cab6a801cc5655_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}}\\frac{\\text{tan}(x)-\\text{sen}(x)}{3x\\cdot\\text{tan}(x)}=\\frac{\\text{tan}(0)-\\text{sen}(0)}{3\\cdot 0\\cdot\\text{tan}(0)}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"334\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u6211\u4eec\u5c06\u6bcf\u4e00\u9879\u9664\u4ee5 x \u7684\u6b63\u5207\u6765\u7b80\u5316\u6781\u9650\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d9036c709f0cf05a0e1d3e53a1f81af8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\displaystyle\\frac{\\text{tan}(x)}{\\text{tan}(x)}-\\frac{\\text{sen}(x)}{\\text{tan}(x)}}{\\displaystyle\\frac{3x\\cdot\\text{tan}(x)}{\\text{tan}(x)}}=\\lim_{x\\to 0}\\frac{\\displaystyle 1-\\frac{\\text{sen}(x)}{\\text{tan}(x)}}{3x}\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"305\" style=\"vertical-align: -40px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5176\u6b21\uff0c\u6211\u4eec\u53ef\u4ee5\u4ece\u57fa\u672c\u4e09\u89d2\u6052\u7b49\u5f0f\u63a8\u5bfc\u51fa\u5206\u5b50\u7684\u5206\u6570\u7b49\u4e8e x \u7684\u4f59\u5f26\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-2c4733e791e3ea6006f69e25c3db9f99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(x)=\\cfrac{\\text{sen}(x)}{\\text{cos}(x)}\\ \\longrightarrow \\ \\text{cos}(x)=\\cfrac{\\text{sen}(x)}{\\text{tan}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"297\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-061c6b3972071af7e6227fad37ec4019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\displaystyle 1-\\frac{\\text{sen}(x)}{\\text{tan}(x)}}{3x}=\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{3x}\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"255\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5e76\u4e14\u5e94\u7528\u4e09\u89d2\u6781\u9650\u7406\u8bba\u4e2d\u6f14\u793a\u7684\u7b2c\u4e8c\u4e2a\u516c\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u8f7b\u677e\u6c42\u89e3\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-51835176b411ca1ec4f37835a83685fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{x}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"146\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b1ee20bc86ad7559fcda3d6bad3c9b27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{3x}=\\lim_{x\\to 0}\\frac{1}{3}\\cdot \\frac{1-\\text{cos}(x)}{x}=\\\\[4ex]\\displaystyle =\\frac{1}{3}\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{x}=\\frac{1}{3}\\cdot 0=\\bm{0}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"101\" width=\"294\" style=\"vertical-align: 0px;\"><\/p>\n<\/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\u4e0b\u5217\u4e09\u89d2\u6781\u9650\u5728\u70b9 x=0 \u5904\u7684\u89e3\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-a7cbd12a8e0f0416e55baa4799395661_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{2\\text{sen}(x)\\text{cos}(x)\\text{sen}(5x)}{x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"196\" 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\">\u5982\u679c\u6211\u4eec\u5c1d\u8bd5\u6c42\u89e3\u6781\u9650\uff0c\u6211\u4eec\u4f1a\u627e\u5230\u4e0d\u5b9a\u5f62\u5f0f 0\/0\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0d1052ddde97caedf5e563febc26fad7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{2\\text{sen}(x)\\text{cos}(x)\\text{sen}(5x)}{x^2}=\\frac{2\\text{sen}(0)\\text{cos}(0)\\text{sen}(5\\cdot 0)}{0^2}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"432\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5206\u5b50\u7684\u4ee3\u6570\u8868\u8fbe\u5f0f\u53ef\u4ee5\u4f7f\u7528\u53cc\u89d2\u6b63\u5f26\u7684\u4e09\u89d2\u6052\u7b49\u5f0f\u91cd\u5199\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-2a7acdc1773c3d7fd430328604cee7d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}(2x)=2\\text{sen}(x)\\text{cos}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"185\" style=\"vertical-align: -5px;\"><\/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-4426a183bebf1739384bda14bcd59dc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{2\\text{sen}(x)\\text{cos}(x)\\text{sen}(5x)}{x^2}=\\lim_{x\\to 0}\\frac{\\text{sen}(2x)\\text{sen}(5x)}{x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"371\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u8ba9\u6211\u4eec\u5c06\u4e09\u89d2\u51fd\u6570\u7684\u6781\u9650\u5206\u89e3\u4e3a\u4e58\u79ef\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-fc650634075435b782f1e7b921b77c02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(2x)\\cdot \\text{sen}(5x)}{x\\cdot x}=\\\\[4ex]\\displaystyle =\\lim_{x\\to 0}\\frac{\\text{sen}(2x)}{x}\\cdot\\frac{\\text{sen}(5x)}{x}=\\\\[4ex]\\displaystyle =\\lim_{x\\to 0}\\frac{\\text{sen}(2x)}{x}\\cdot\\lim_{x\\to 0}\\frac{\\text{sen}(5x)}{x}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"163\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u901a\u8fc7\u5e94\u7528\u6781\u9650\u7684\u6027\u8d28\u6765\u6c42\u89e3\u4e09\u89d2\u6781\u9650\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7c26ba3032828541e69e4bd976ac4f96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(2x)}{x}\\cdot\\lim_{x\\to 0}\\frac{\\text{sen}(5x)}{x}=\\\\[4ex]\\displaystyle =2\\cdot \\lim_{x\\to 0}\\frac{\\text{sen}(2x)}{2x}\\cdot 5\\cdot \\lim_{x\\to 0}\\frac{\\text{sen}(5x)}{5x}=\\\\[4ex]\\displaystyle =2\\cdot 1\\cdot 5\\cdot 1=\\bm{10}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"141\" width=\"278\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u5728\u8fd9\u91cc\u60a8\u5c06\u4e86\u89e3\u5982\u4f55\u6c42\u89e3\u4e09\u89d2\u6781\u9650\u3002\u60a8\u5c06\u80fd\u591f\u770b\u5230\u4e09\u89d2\u51fd\u6570\u6781\u9650\u7684\u51e0\u4e2a\u793a\u4f8b\uff0c\u751a\u81f3\u53ef\u4ee5\u901a\u8fc7\u89e3\u51b3\u4e09\u89d2\u51fd\u6570\u6781\u9650\u7684\u5206\u6b65\u7ec3\u4e60\u8fdb\u884c &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/cn\/%e4%b8%89%e8%a7%92%e6%9e%81%e9%99%90\/\"> <span class=\"screen-reader-text\">\u4e09\u89d2\u6781\u9650<\/span> 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