{"id":3958,"date":"2021-08-11T12:52:23","date_gmt":"2021-08-11T12:52:23","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3958"},"modified":"2021-11-26T16:37:56","modified_gmt":"2021-11-26T11:07:56","slug":"limit-of-trigonometric-functions","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/","title":{"rendered":"How to Find Limit of Trigonometric Functions"},"content":{"rendered":"

Here, you will learn how to find limit of trigonometric functions and limits using series expansion with example.<\/p>\n

Let’s begin –\u00a0<\/p>\n

Limit of Trigonometric Functions<\/h2>\n

\\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(sinx\\over x\\) = 1 = \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(tanx\\over x\\) = \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(tan^{-1}x\\over x\\) = \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(sin^{-1}x\\over x\\) [where x is measured in radians]<\/p>\n

(a)\u00a0 If \\(\\displaystyle{\\lim_{x \\to a}}\\) f(x) = 0, then \\(\\displaystyle{\\lim_{x \\to a}}\\) \\(sinf(x)\\over f(x)\\) = 1
e.g. \\(\\displaystyle{\\lim_{x \\to 1}}\\) \\(sin(lnx)\\over (lnx)\\) = 1\u00a0<\/p>\n\n\n

Example : <\/span>Evaluate : \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(x^3 cotx\\over {1-cosx}\\)<\/p>\n

Solution : <\/span>\n \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(x^3 cosx\\over {sinx(1-cosx)}\\) = \n\t\t \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(x^3 cosx(1 + cosx)\\over {sinxsin^2x}\\) = \n\t\t \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\({x^3\\over sin^3x}.cosx(1 + cosx)\\) = 2

<\/p>\n\n\n\n

Example : <\/span> Evaluate : \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\((2+x)sin(2+x)-2sin2\\over x\\)<\/p>\n

Solution : <\/span>\\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(2(sin(2+x)-sin2)+xsin(2+x)\\over x\\)

\n = \\(\\displaystyle{\\lim_{x \\to 0}}\\)(\\(2.2.cos(2+{x\\over 2})sin{x\\over 2}\\over x\\) + sin(2+x))

= \\(\\displaystyle{\\lim_{x \\to 0}}\\)\\(2cos(2+{x\\over 2})sin{x\\over 2}\\over {x\\over 2}\\) + \\(\\displaystyle{\\lim_{x \\to 0}}\\)sin(2+x)

= 2cos2 + sin2

<\/p>\n\n\n\n

Example : <\/span>Evaluate : \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(xln(1+2tanx)\\over 1-cosx\\)<\/p>\n

Solution : <\/span>\\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(xln(1+2tanx)\\over 1-cosx\\)

= \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(xln(1+2tanx)\\over {1-cosx\\over x^2}.x^2\\).\\(2tanx\\over 2tanx\\)

= 4

<\/p>\n\n\n

Limit using series expansion<\/h2>\n

Expansion of function like binomial expansion, exponential & logarithmic expansion, expansion of sinx, cosx, tanx should be remembered by heart which are given below :<\/p>\n

(a)  \\(e^x\\) = 1 + \\(x\\over 1!\\) + \\(x^2\\over {2!}\\) + ……..<\/p>\n

(b)  ln(1 + x) = x – \\(x^2\\over 2\\) + \\(x^3\\over 3\\) – \\(x^4\\over 4\\) + ………for -1 < x \\(\\leq\\) 1<\/p>\n

(c)  sinx = x – \\(x^3\\over 3!\\) + \\(x^5\\over 5!\\) – \\(x^7\\over 7!\\) + ……….<\/p>\n

(d)  cosx = 1 – \\(x^2\\over 2!\\) + \\(x^4\\over 4!\\) + \\(x^6\\over 6!\\) + ……….<\/p>\n

(e)  tanx = x + \\(x^3\\over 3\\) + \\(2x^5\\over 15\\) + …….<\/p>\n

Hope you learnt how to find the limit of trigonometric functions. To learn more practice more questions and get ahead in competition. Good Luck!<\/p>\n\n\n

\n
Next – What is squeeze theorem – Limit of exponential functions<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – How to solve Indeterminate forms of limits<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn how to find limit of trigonometric functions and limits using series expansion with example. Let’s begin –\u00a0 Limit of Trigonometric Functions \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(sinx\\over x\\) = 1 = \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(tanx\\over x\\) = \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(tan^{-1}x\\over x\\) = \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(sin^{-1}x\\over x\\) [where x is measured in …<\/p>\n

How to Find Limit of Trigonometric Functions<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[19],"tags":[451,452],"yoast_head":"\nHow to Find Limit of Trigonometric Functions - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn how to find limit of trigonometric functions and limits using series expansion with example.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How to Find Limit of Trigonometric Functions - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"In this post, you will learn how to find limit of trigonometric functions and limits using series expansion with example.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-08-11T12:52:23+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-26T11:07:56+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"How to Find Limit of Trigonometric Functions\",\"datePublished\":\"2021-08-11T12:52:23+00:00\",\"dateModified\":\"2021-11-26T11:07:56+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/\"},\"wordCount\":361,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"limit of trigonometric functions\",\"limit using series expansion\"],\"articleSection\":[\"Limits\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/\",\"url\":\"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/\",\"name\":\"How to Find Limit of Trigonometric Functions - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-08-11T12:52:23+00:00\",\"dateModified\":\"2021-11-26T11:07:56+00:00\",\"description\":\"In this post, you will learn how to find limit of trigonometric functions and limits using series expansion with example.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"How to Find Limit of Trigonometric Functions\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - Study Math Online\",\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathemerize.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathemerize.com\/#organization\",\"name\":\"Mathemerize\",\"url\":\"https:\/\/mathemerize.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"width\":140,\"height\":96,\"caption\":\"Mathemerize\"},\"image\":{\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.instagram.com\/mathemerize\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\",\"name\":\"mathemerize\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"caption\":\"mathemerize\"},\"sameAs\":[\"https:\/\/mathemerize.com\"],\"url\":\"https:\/\/mathemerize.com\/author\/mathemerize\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"How to Find Limit of Trigonometric Functions - Mathemerize","description":"In this post, you will learn how to find limit of trigonometric functions and limits using series expansion with example.","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:\/\/mathemerize.com\/limit-of-trigonometric-functions\/","og_locale":"en_US","og_type":"article","og_title":"How to Find Limit of Trigonometric Functions - Mathemerize","og_description":"In this post, you will learn how to find limit of trigonometric functions and limits using series expansion with example.","og_url":"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/","og_site_name":"Mathemerize","article_published_time":"2021-08-11T12:52:23+00:00","article_modified_time":"2021-11-26T11:07:56+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"How to Find Limit of Trigonometric Functions","datePublished":"2021-08-11T12:52:23+00:00","dateModified":"2021-11-26T11:07:56+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/"},"wordCount":361,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"keywords":["limit of trigonometric functions","limit using series expansion"],"articleSection":["Limits"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/","url":"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/","name":"How to Find Limit of Trigonometric Functions - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-08-11T12:52:23+00:00","dateModified":"2021-11-26T11:07:56+00:00","description":"In this post, you will learn how to find limit of trigonometric functions and limits using series expansion with example.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/limit-of-trigonometric-functions\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"How to Find Limit of Trigonometric Functions"}]},{"@type":"WebSite","@id":"https:\/\/mathemerize.com\/#website","url":"https:\/\/mathemerize.com\/","name":"Mathemerize","description":"Maths Tutorials - Study Math Online","publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathemerize.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/mathemerize.com\/#organization","name":"Mathemerize","url":"https:\/\/mathemerize.com\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/","url":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","contentUrl":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","width":140,"height":96,"caption":"Mathemerize"},"image":{"@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.instagram.com\/mathemerize\/"]},{"@type":"Person","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df","name":"mathemerize","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","caption":"mathemerize"},"sameAs":["https:\/\/mathemerize.com"],"url":"https:\/\/mathemerize.com\/author\/mathemerize\/"}]}},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/3958"}],"collection":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/comments?post=3958"}],"version-history":[{"count":5,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/3958\/revisions"}],"predecessor-version":[{"id":8665,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/3958\/revisions\/8665"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=3958"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=3958"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=3958"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}