{"id":13947,"date":"2020-12-10T11:25:12","date_gmt":"2020-12-10T05:55:12","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=13947"},"modified":"2020-12-10T15:38:26","modified_gmt":"2020-12-10T10:08:26","slug":"evaluating-limits","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/evaluating-limits\/","title":{"rendered":"Evaluating Limits"},"content":{"rendered":"<h2>Evaluating Limits<\/h2>\n<p><iframe loading=\"lazy\" title=\"Solving limits by factoring - Calculus\" width=\"1200\" height=\"900\" src=\"https:\/\/www.youtube.com\/embed\/2EpT-zZe3dA?start=155&#038;feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/p>\n<h3>Methods of evaluation of limits<\/h3>\n<p>We shall divide the problems of evaluation of limits in five categories.<\/p>\n<h3>(1) Algebraic limits:<\/h3>\n<p>Let f(x) be an algebraic function and \u2018a\u2019 be a real number. Then <img loading=\"lazy\" class=\"alignnone size-full wp-image-129700\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-1.jpg\" alt=\"Evaluating Limits 1\" width=\"80\" height=\"41\" \/>\u00a0is known as an algebraic limit.<\/p>\n<ol>\n<li><strong>Direct substitution method:<\/strong> If by direct substitution of the point in the given expression we get a finite number, then the number obtained is the limit of the given expression.<\/li>\n<li><strong>Factorisation method:<\/strong> In this method, numerator and denominator are factorised. The common factors are cancelled and the rest outputs the results.<\/li>\n<li><strong>Rationalisation method:<\/strong> Rationalisation is followed when we have fractional powers (like \\(\\frac { 1 }{ 2 } ,\\frac { 1 }{ 3 }\\)\u00a0etc.) on expressions in numerator or denominator or in both. After rationalisation the terms are factorised which on cancellation gives the result.<\/li>\n<li><strong>Based on the form when x \u2192 \u221e:<\/strong> In this case expression should be expressed as a function 1\/x and then after removing indeterminate form, (if it is there) replace 1\/x by 0.<\/li>\n<\/ol>\n<h3>(2) Trigonometric limits:<\/h3>\n<p>To evaluate trigonometric limit the following results are very important.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129702\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-2.jpg\" alt=\"Evaluating Limits 2\" width=\"420\" height=\"796\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-2.jpg 420w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-2-158x300.jpg 158w\" sizes=\"(max-width: 420px) 100vw, 420px\" \/><\/p>\n<h3>(3) Logarithmic limits:<\/h3>\n<p>To evaluate the logarithmic limits we use following formulae:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129703\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-3.jpg\" alt=\"Evaluating Limits 3\" width=\"627\" height=\"327\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-3.jpg 627w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-3-300x156.jpg 300w\" sizes=\"(max-width: 627px) 100vw, 627px\" \/><\/p>\n<h3>(4) Exponential limits:<\/h3>\n<p><strong>(i) Based on series expansion:<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129704\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-4.jpg\" alt=\"Evaluating Limits 4\" width=\"422\" height=\"66\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-4.jpg 422w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-4-300x47.jpg 300w\" sizes=\"(max-width: 422px) 100vw, 422px\" \/><br \/>\nTo evaluate the exponential limits we use the following results:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129705\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-5.jpg\" alt=\"Evaluating Limits 5\" width=\"414\" height=\"201\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-5.jpg 414w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-5-300x146.jpg 300w\" sizes=\"(max-width: 414px) 100vw, 414px\" \/><br \/>\n<strong>(ii) Based on the form 1<sup>\u221e<\/sup>:<\/strong> To evaluate the exponential form <strong>1<sup>\u221e<\/sup><\/strong>\u00a0we use the following results.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129706\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-6.jpg\" alt=\"Evaluating Limits 6\" width=\"631\" height=\"502\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-6.jpg 631w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-6-300x239.jpg 300w\" sizes=\"(max-width: 631px) 100vw, 631px\" \/><\/p>\n<h3>(5) L-Hospital\u2019s rule:<\/h3>\n<p><iframe loading=\"lazy\" title=\"Using L&#039;Hopital&#039;s Rule\" width=\"1200\" height=\"900\" src=\"https:\/\/www.youtube.com\/embed\/W5HYPtriZ7M?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/p>\n<p>If f(x) and g(x) be two functions of x such that<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129707\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-7.jpg\" alt=\"Evaluating Limits 7\" width=\"633\" height=\"491\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-7.jpg 633w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-7-300x233.jpg 300w\" sizes=\"(max-width: 633px) 100vw, 633px\" \/><br \/>\nSometimes it may be necessary to repeat this process a number of times till our goal of evaluating limit is achieved.<\/p>\n<h3>Evaluating Limits Problems with Solutions<\/h3>\n<p><strong>1.<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129709\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-8.jpg\" alt=\"Evaluating Limits 8\" width=\"492\" height=\"137\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-8.jpg 492w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-8-300x84.jpg 300w\" sizes=\"(max-width: 492px) 100vw, 492px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129710\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-9.jpg\" alt=\"Evaluating Limits 9\" width=\"542\" height=\"372\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-9.jpg 542w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-9-300x206.jpg 300w\" sizes=\"(max-width: 542px) 100vw, 542px\" \/><br \/>\n<strong>2.<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129712\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-10.jpg\" alt=\"Evaluating Limits 10\" width=\"498\" height=\"134\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-10.jpg 498w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-10-300x81.jpg 300w\" sizes=\"(max-width: 498px) 100vw, 498px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129714\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-11.jpg\" alt=\"Evaluating Limits 11\" width=\"564\" height=\"180\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-11.jpg 564w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-11-300x96.jpg 300w\" sizes=\"(max-width: 564px) 100vw, 564px\" \/><br \/>\n<strong>3.<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129715\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-12.jpg\" alt=\"Evaluating Limits 12\" width=\"471\" height=\"129\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-12.jpg 471w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-12-300x82.jpg 300w\" sizes=\"(max-width: 471px) 100vw, 471px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129716\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-13.jpg\" alt=\"Evaluating Limits 13\" width=\"546\" height=\"245\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-13.jpg 546w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-13-300x135.jpg 300w\" sizes=\"(max-width: 546px) 100vw, 546px\" \/><br \/>\n<strong>4.<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129717\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-14.jpg\" alt=\"Evaluating Limits 14\" width=\"477\" height=\"124\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-14.jpg 477w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-14-300x78.jpg 300w\" sizes=\"(max-width: 477px) 100vw, 477px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129718\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-15.jpg\" alt=\"Evaluating Limits 15\" width=\"549\" height=\"343\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-15.jpg 549w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-15-300x187.jpg 300w\" sizes=\"(max-width: 549px) 100vw, 549px\" \/><br \/>\n<strong>5.<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129727\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-16.jpg\" alt=\"Evaluating Limits 16\" width=\"528\" height=\"123\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-16.jpg 528w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-16-300x70.jpg 300w\" sizes=\"(max-width: 528px) 100vw, 528px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129741\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-17.jpg\" alt=\"Evaluating Limits 17\" width=\"565\" height=\"91\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-17.jpg 565w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-17-300x48.jpg 300w\" sizes=\"(max-width: 565px) 100vw, 565px\" \/><br \/>\n<strong>6.<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129742\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-18.jpg\" alt=\"Evaluating Limits 18\" width=\"516\" height=\"164\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-18.jpg 516w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-18-300x95.jpg 300w\" sizes=\"(max-width: 516px) 100vw, 516px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129744\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-19.jpg\" alt=\"Evaluating Limits 19\" width=\"558\" height=\"414\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-19.jpg 558w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-19-300x223.jpg 300w\" sizes=\"(max-width: 558px) 100vw, 558px\" \/><br \/>\n<strong>7.<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129745\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-20.jpg\" alt=\"Evaluating Limits 20\" width=\"463\" height=\"127\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-20.jpg 463w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-20-300x82.jpg 300w\" sizes=\"(max-width: 463px) 100vw, 463px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129746\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-21.jpg\" alt=\"Evaluating Limits 21\" width=\"535\" height=\"223\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-21.jpg 535w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-21-300x125.jpg 300w\" sizes=\"(max-width: 535px) 100vw, 535px\" \/><br \/>\n<strong>8.<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129747\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-22.jpg\" alt=\"Evaluating Limits 22\" width=\"538\" height=\"167\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-22.jpg 538w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-22-300x93.jpg 300w\" sizes=\"(max-width: 538px) 100vw, 538px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-129748\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-23.jpg\" alt=\"Evaluating Limits 23\" width=\"547\" height=\"354\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-23.jpg 547w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Evaluating-Limits-23-300x194.jpg 300w\" sizes=\"(max-width: 547px) 100vw, 547px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Evaluating Limits Methods of evaluation of limits We shall divide the problems of evaluation of limits in five categories. (1) Algebraic limits: Let f(x) be an algebraic function and \u2018a\u2019 be a real number. Then \u00a0is known as an algebraic limit. Direct substitution method: If by direct substitution of the point in the given expression &#8230; <a title=\"Evaluating Limits\" class=\"read-more\" href=\"https:\/\/cbselibrary.com\/evaluating-limits\/\" aria-label=\"More on Evaluating Limits\">Read more<\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[5257,5261,5258,5255,5266,5264,5259,5265,5263,5256,5260,5262],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v18.4.1) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Evaluating Limits - CBSE Library<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/cbselibrary.com\/evaluating-limits\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Evaluating Limits\" \/>\n<meta property=\"og:description\" content=\"Evaluating Limits Methods of evaluation of limits We shall divide the problems of evaluation of limits in five categories. 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CBSE Library","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/cbselibrary.com\/evaluating-limits\/","og_locale":"en_US","og_type":"article","og_title":"Evaluating Limits","og_description":"Evaluating Limits Methods of evaluation of limits We shall divide the problems of evaluation of limits in five categories. (1) Algebraic limits: Let f(x) be an algebraic function and \u2018a\u2019 be a real number. Then \u00a0is known as an algebraic limit. Direct substitution method: If by direct substitution of the point in the given expression ... 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