{"id":9568,"date":"2018-02-13T10:09:37","date_gmt":"2018-02-13T04:39:37","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/?p=9568"},"modified":"2018-10-09T11:39:56","modified_gmt":"2018-10-09T06:09:56","slug":"limits-and-derivatives-class-11-notes-mathematics","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/limits-and-derivatives-class-11-notes-mathematics\/","title":{"rendered":"Limits And Derivatives class 11 Notes Mathematics"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/limits-and-derivatives-class-11-notes-mathematics\/#CBSE_Guide_Limits_And_Derivatives_class_11_Maths_Notes\" >CBSE Guide Limits And Derivatives class 11 Maths Notes<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/limits-and-derivatives-class-11-notes-mathematics\/#Limits_And_Derivatives_class_11_Notes_Mathematics\" >Limits And Derivatives class 11 Notes Mathematics<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/limits-and-derivatives-class-11-notes-mathematics\/#CBSE_Class_11_Mathematics_Revision_Notes_Chapter_13_Limits_And_Derivatives\" >CBSE Class 11 Mathematics Revision Notes Chapter 13 Limits And Derivatives<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/limits-and-derivatives-class-11-notes-mathematics\/#Following_are_some_of_the_standard_limits\" >Following are some of the standard limits<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/limits-and-derivatives-class-11-notes-mathematics\/#Derivatives\" >Derivatives<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/limits-and-derivatives-class-11-notes-mathematics\/#Following_are_some_of_the_standard_derivatives\" >Following are some of the standard derivatives<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/limits-and-derivatives-class-11-notes-mathematics\/#Limits_And_Derivatives_class_11_Notes\" >Limits And Derivatives class 11 Notes<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/limits-and-derivatives-class-11-notes-mathematics\/#CBSE_Class_11_Revision_Notes_and_Key_Points\" >CBSE Class 11 Revision Notes and Key Points<\/a><\/li><\/ul><\/nav><\/div>\n<p>CBSE Mathematics Chapter 13 Limits And Derivatives class 11 Notes Mathematics in PDF are available for free download in myCBSEguide mobile app. The best app for CBSE students now provides Limits And Derivatives class 11 Notes Maths latest chapter wise notes for quick preparation of CBSE exams and school based annual examinations. Class 11 Mathematics notes on Chapter 13 Limits And Derivatives class 11 Notes Mathematics are also available for download in CBSE Guide website.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_Guide_Limits_And_Derivatives_class_11_Maths_Notes\"><\/span>CBSE Guide Limits And Derivatives class 11 Maths Notes<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>CBSE guide notes are the comprehensive notes which covers the latest syllabus of CBSE and NCERT. It includes all the topics given in NCERT class 11 Mathematics text book. Users can download CBSE guide quick revision notes from myCBSEguide mobile app and my CBSE guide website.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Limits_And_Derivatives_class_11_Notes_Mathematics\"><\/span>Limits And Derivatives class 11 Notes Mathematics<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Download CBSE class 11th revision notes for Chapter 13 Limits And Derivatives class 11 Notes Mathematics in PDF format for free. Download revision notes for Limits And Derivatives class 11 Notes Mathematics and score high in exams. These are the Limits And Derivatives class 11 Notes Mathematics prepared by team of expert teachers. The revision notes help you revise the whole chapter in minutes. Revising notes in exam days is on of the best tips recommended by teachers during exam days.<\/p>\n<p style=\"text-align: center;\"><strong>NCERT solutions for Class 11 Maths\u00a0<a class=\"button\" href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-11-mathematics\/1371\/cbse-revision-notes\/7\/\">Download as PDF<\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_Class_11_Mathematics_Revision_Notes_Chapter_13_Limits_And_Derivatives\"><\/span><strong>CBSE Class 11 Mathematics Revision Notes Chapter 13 Limits And Derivatives<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ol>\n<li><strong>Limits <\/strong><\/li>\n<li><strong>Derivatives <\/strong><\/li>\n<li><strong>Miscellaneous Questions<\/strong><\/li>\n<\/ol>\n<p><strong>Meaning of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x \\rightarrow a\" \/><\/span><\/span>\u00a0or &#8220;<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x\" \/><\/span><\/span>\u00a0tends to <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a\" \/><\/span><\/span>&#8221; or &#8220;<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x\" \/><\/span><\/span>\u00a0approaches <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a\" \/><\/span><\/span><\/strong>&#8220;, <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x\" \/><\/span><\/span>\u00a0<strong>is a variable<\/strong>. The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit. It can be changed so that its value comes nearer and nearer to <strong><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a\" \/><\/span><\/span><\/strong>.\u00a0 <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?0 &amp;lt; \\left| {x - a} \\right| &amp;lt; \\delta\" \/><\/span><\/span><\/p>\n<p>(i) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x \\ne a\" \/><\/span><\/span>,\u00a0\u00a0\u00a0\u00a0\u00a0 (ii) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\left| {x - a} \\right|\" \/><\/span><\/span>\u00a0becomes smaller and smaller as we please.<\/p>\n<p><strong>Neighbourhood<\/strong>: The set of all real numbers lying between <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a - \\delta\" \/><\/span><\/span>\u00a0and <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a + \\delta\" \/><\/span><\/span>\u00a0is called the neighbourhood of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a\" \/><\/span><\/span>. Neighbourhood of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a\" \/><\/span><\/span>\u00a0= <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\left( {a - \\delta ,a + \\delta } \\right),\" \/><\/span><\/span>\u00a0 \u00a0 \u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x \\in \\left( {a - \\delta ,a + \\delta } \\right)\" \/><\/span><\/span><\/p>\n<ul>\n<li><strong>Limit <\/strong>of a function at a point is the common value of the left and right hand limits, if they coincide<\/li>\n<\/ul>\n<p><strong>Left hand limit of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0at <\/strong><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = a\" \/><\/span><\/span>. When <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x\" \/><\/span><\/span>\u00a0approches <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a\" \/><\/span><\/span>\u00a0from left hand side of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a\" \/><\/span><\/span>, the function <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f(x)\" \/><\/span><\/span>\u00a0tends to <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?l\" \/><\/span><\/span>\u00a0&#8220;a definite number&#8221;. This definite number <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?l\" \/><\/span><\/span>\u00a0is said to be the left hand limit of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0at <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = a\" \/><\/span><\/span>.<\/p>\n<p><b>Right hand limit of <\/b><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0at <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = a\" \/><\/span><\/span>. When <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x\" \/><\/span><\/span>\u00a0approches <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a\" \/><\/span><\/span>\u00a0from\u00a0right hand side of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a\" \/><\/span><\/span>, the function <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f(x)\" \/><\/span><\/span>\u00a0tends to <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?l\" \/><\/span><\/span>\u00a0&#8220;a definite number&#8221;. This definite number <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?l\" \/><\/span><\/span>\u00a0is said to be the\u00a0right hand limit of \u00a0at <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = a\" \/><\/span><\/span>.<\/p>\n<p>Therefore, if Left hand limit of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0at <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = a\" \/><\/span><\/span>\u00a0= Right hand limit of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0at <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = a\" \/><\/span><\/span>, then the limit of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f(x)\" \/><\/span><\/span>\u00a0at <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = a\" \/><\/span><\/span>\u00a0exists.<\/p>\n<ul>\n<li>For function f and a real number a, <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\underset{x\\to \\infty }{\\mathop{\\lim }}\\,f(x)\" \/><\/span><\/span>\u00a0and f (a) may not be same (Infact, one may be defined and not the other one).<\/li>\n<li>For functions f and g the following holds:.<\/li>\n<\/ul>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\mathop {\\lim }\\limits_{x \\to a} \\left[ {f(x) \\pm g(x)} \\right] =\" \/><\/span><\/span><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\mathop {\\lim }\\limits_{x \\to a} f(x) \\pm \\mathop {\\lim }\\limits_{x \\to a} g(x)\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\underset{x\\to a}{\\mathop{\\lim }}\\,\\left[ f(x).g(x) \\right]=\" \/><\/span><\/span><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\underset{x\\to a}{\\mathop{\\lim }}\\,f(x).\\underset{x\\to a}{\\mathop{\\lim }}\\,g(x)\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\underset{x\\to a}{\\mathop{\\lim }}\\,\\left[ \\frac{f(x)}{g(x)} \\right]=\" \/><\/span><\/span><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\frac{\\underset{x\\to a}{\\mathop{\\lim }}\\,f(x)}{\\underset{x\\to \\infty }{\\mathop{\\lim }}\\,g(x)}\" \/><\/span><\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Following_are_some_of_the_standard_limits\"><\/span>Following are some of the standard limits<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\underset{x\\to a}{\\mathop{\\lim }}\\,\\frac{{{x}^{n}}-{{a}^{n}}}{x-a}=n{{a}^{n-1}}\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\underset{x\\to 0}{\\mathop{\\lim }}\\,\\frac{\\sin x}{x}=1\" \/><\/span><\/span>, <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\mathop {\\lim }\\limits_{x \\to a} {{\\sin \\left( {x - a} \\right)} \\over {x - a}} = 1\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\underset{x\\to 0}{\\mathop{\\lim }}\\,\\frac{1-\\cos x}{x}=0\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\mathop {\\lim }\\limits_{x \\to 0} {{\\tan x} \\over x} = 1,\" \/><\/span><\/span><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\mathop {\\lim }\\limits_{x \\to a} {{\\tan \\left( {x - a} \\right)} \\over {x - a}} = 1\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\mathop {\\lim }\\limits_{x \\to 0} {{{{\\sin }^{ - 1}}x} \\over x} = 1,\" \/><\/span><\/span><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\mathop {\\lim }\\limits_{x \\to 0} {{{{\\tan }^{ - 1}}x} \\over x} = 1\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\mathop {\\lim }\\limits_{x \\to 0} {{{a^x} - 1} \\over x} = {\\log _e}a,\" \/><\/span><\/span><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a &amp;gt; 0,a \\ne 1\" \/><\/span><\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Derivatives\"><\/span><strong>Derivatives<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The derivative of a function f at a is defined by<\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f'(a)=\\underset{h\\to 0}{\\mathop{\\lim }}\\,\\frac{f(a+h)-f(a)}{h}\" \/><\/span><\/span><\/p>\n<p>Derivative of a function f at any point x is defined by<\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f'(x)=\\frac{df(x)}{dx}=\\underset{h\\to 0}{\\mathop{\\lim }}\\,\\frac{f(x+h)-f(x)}{h}\" \/><\/span><\/span><\/p>\n<p>For functions u and v the following holds:<\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?(u\\pm v)'=u'\\pm v'\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?(uv)'=u'v+uv'\" \/><\/span><\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\Rightarrow\" \/><\/span><\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {uv} \\right) = u.{{dv} \\over {dx}} + v.{{du} \\over {dx}}\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{\\left( \\frac{u}{v} \\right)}^{'}}=\\frac{u'v-uv'}{{{v}^{2}}}\" \/><\/span><\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\Rightarrow\" \/><\/span><\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {{u \\over v}} \\right) = {{v.{{du} \\over {dx}} - u.{{dv} \\over {dx}}} \\over {{v^2}}}\" \/><\/span><\/span><\/p>\n<p>provided all are defind.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Following_are_some_of_the_standard_derivatives\"><\/span>Following are some of the standard derivatives<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\frac{d}{dx}({{x}^{n}})=n{{x}^{n-1}}\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\frac{d}{dx}(\\sin x)=\\cos x\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\frac{d}{dx}(\\cos x)=-\\sin x\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {\\tan x} \\right) = {\\sec ^2}x\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {\\cot x} \\right) = - \\cos e{c^2}x\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {\\sec x} \\right) = \\sec x.\\tan x\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {\\cos ecx} \\right) = - \\cos ec{\\rm{ }}x.\\cot x\" \/><\/span><\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Limits_And_Derivatives_class_11_Notes\"><\/span>Limits And Derivatives class 11 Notes<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>CBSE Revision notes (PDF Download) Free<\/li>\n<li>CBSE Revision notes for Class 11 Mathematics PDF<\/li>\n<li>Revision notes Class 11 Maths \u2013 CBSE<\/li>\n<li>CBSE Revisions notes and Key Points Class 11 Mathematics<\/li>\n<li>Summary of the NCERT books all chapters in Mathematics class 11<\/li>\n<li>Short notes for CBSE class 11th Mathematics<\/li>\n<li>Key notes and chapter summary of Mathematics class 11<\/li>\n<li>Quick revision notes for CBSE exams<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_Class_11_Revision_Notes_and_Key_Points\"><\/span><strong>CBSE Class 11 Revision Notes and Key Points<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Limits And Derivatives class 11 Notes Mathematics. CBSE quick revision note for class-11 Mathematics, Physics, Chemistry, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.<\/p>\n<ul>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-11-physics\/1340\/cbse-revision-notes\/7\/\">Physics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-11-chemistry\/1356\/cbse-revision-notes\/7\/\">Chemistry<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-11-mathematics\/1371\/cbse-revision-notes\/7\/\">Mathematics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-11-biology\/1388\/cbse-revision-notes\/7\/\">Biology<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-11-accountancy\/1411\/cbse-revision-notes\/7\/\">Accountancy<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-11-economics\/1423\/cbse-revision-notes\/7\/\">Economics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-11-business-studies\/1740\/cbse-revision-notes\/7\/\">Business Studies<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-11-computer-science\/1852\/cbse-revision-notes\/7\/\"> Computer Science<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-11-informatics-practices\/1874\/cbse-revision-notes\/7\/\">Informatics Practices<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-11-geography\/1864\/cbse-revision-notes\/7\/\">Geography<\/a><\/li>\n<\/ul>\n<p>To download Limits And Derivatives class 11 Notes, sample paper for class 11 Chemistry, Physics, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n<ul>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/sets-class-11-notes-mathematics\/\">Sets class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/relations-functions-class-11-notes-mathematics\/\">Relations And Functions class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/trigonometric-functions-class-11-notes-mathematics\/\">Trigonometric Functions class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/principles-mathematical-induction-class-11-notes-mathematics\/\">Principles Of Mathematical Induction class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/complex-numbers-and-quadratic-equations-class-11-notes-mathematics\/\">Complex Numbers And Quadratic Equations class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/linear-inequalities-class-11-notes-mathematics\/\">Linear Inequalities class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/permutations-combinations-class-11-notes-mathematics\/\">Permutations And Combinations class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/binomial-theorem-class-11-notes-mathematics\/\">Binomial Theorem class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/sequences-and-series-class-11-notes-mathematics\/\">Sequences And Series class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/straight-lines-class-11-notes-mathematics\/\">Straight Lines class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/conic-sections-class-11-notes-mathematics\/\">Conic Sections class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/mathematical-reasoning-class-11-notes-mathematics\/\">Mathematical Reasoning class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/statistics-class-11-notes-mathematics\/\">Statistics class 11 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/probability-class-11-notes-mathematics\/\">Probability class 11 Notes Mathematics<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>CBSE Mathematics Chapter 13 Limits And Derivatives class 11 Notes Mathematics in PDF are available for free download in myCBSEguide mobile app. The best app for CBSE students now provides Limits And Derivatives class 11 Notes Maths latest chapter wise notes for quick preparation of CBSE exams and school based annual examinations. Class 11 Mathematics &#8230; <a title=\"Limits And Derivatives class 11 Notes Mathematics\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/limits-and-derivatives-class-11-notes-mathematics\/\" aria-label=\"More on Limits And Derivatives class 11 Notes Mathematics\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":9298,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[47,456],"tags":[457,150,656,461,426,240],"class_list":["post-9568","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-cbse-class-11","category-revision-notes","tag-cbse-notes","tag-cbse-notes-and-key-points","tag-limits-and-derivatives","tag-mathematics-notes","tag-quick-revision","tag-quick-revision-notes"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Limits And Derivatives class 11 Notes Mathematics<\/title>\n<meta name=\"description\" content=\"Limits And Derivatives class 11 Notes Mathematics Chapter 13 in PDF format for free download. 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