{"id":3961,"date":"2021-08-11T13:12:22","date_gmt":"2021-08-11T13:12:22","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3961"},"modified":"2021-11-26T16:35:56","modified_gmt":"2021-11-26T11:05:56","slug":"what-is-squeeze-theorem","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/","title":{"rendered":"What is Squeeze Theorem – Limit of Exponential Functions"},"content":{"rendered":"

Here, you will learn what is squeeze theorem or sandwich theorem of limit and limit of exponential function with examples.<\/p>\n

Let’s begin –<\/p>\n

Squeeze Theorem (Sandwich Theorem)<\/h2>\n
\n

If f(x) \\(\\leq\\) g(x) \\(\\leq\\) h(x); \\(\\forall\\) x in the neighbourhood at x = a and<\/p>\n

\\(\\displaystyle{\\lim_{x \\to a}}\\) f(x) = l = \\(\\displaystyle{\\lim_{x \\to 1}}\\) h(x) then \\(\\displaystyle{\\lim_{x \\to 1}}\\) g(x) = l<\/p>\n<\/blockquote>\n\n\n

Example : <\/span>Evaluate : \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(x^2sin {1\\over x}\\) = 0 <\/p>\n

Solution : <\/span>\n \\(sin ({1\\over x})\\) lies between -1 and 1.

\n\t\t \\(\\implies\\) \\(-x^2\\) \\(\\leq\\) \\(x^2sin {1\\over x}\\) \\(\\leq\\) \\(x^2\\)

\n\t\t \\(\\implies\\) \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(x^2sin {1\\over x}\\) = 0 as \n\t\t \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(-x^2\\) = \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(x^2\\) = 0

<\/p>\n\n\n

Limit of Exponential Functions<\/h2>\n

(a)  \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(a^x – 1\\over x\\) = lna (a > 0) In particular \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(e^x – 1\\over x\\) = 1<\/p>\n

In general if \\(\\displaystyle{\\lim_{x \\to a}}\\) f(x) = 0, then \\(\\displaystyle{\\lim_{x \\to a}}\\) \\(a^{f(x) – 1}\\over {f(x)}\\) = lna, a > 0<\/p>\n

(b)  (i)  \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\({(1 + x)}^{1\\over x}\\) = e = \\(\\displaystyle{\\lim_{x \\to \\infty}}\\) \\({(1 + {1\\over x})}^x\\) (The base and exponent depends on the same variable.)<\/p>\n

In general if \\(\\displaystyle{\\lim_{x \\to a}}\\) f(x) = 0, then \\(\\displaystyle{\\lim_{x \\to a}}\\) \\({(1 + f(x))}^{1\/f(x)}\\) = e<\/p>\n

(ii)  \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(ln(1 + x)\\over x\\) = 1<\/p>\n

(iii)  If \\(\\displaystyle{\\lim_{x \\to a}}\\) f(x) = 1 and \\(\\displaystyle{\\lim_{x \\to a}}\\) \\(\\phi(x)\\) = \\(\\infty\\) then; \\(\\displaystyle{\\lim_{x \\to a}}\\) \\({[f(x)]}^{\\phi(x)}\\) = \\(e^k\\); where k = \\(\\displaystyle{\\lim_{x \\to a}}\\) \\(\\phi(x)\\) [f(x) – 1]<\/p>\n

(c)  If \\(\\displaystyle{\\lim_{x \\to a}}\\) f(x) = A > 0 & \\(\\displaystyle{\\lim_{x \\to a}}\\) \\(\\phi(x)\\) = B, then \\(\\displaystyle{\\lim_{x \\to a}}\\) \\({[f(x)]}^{\\phi(x)}\\) = \\(e^{B lnA}\\) = \\(A^B\\)<\/p>\n\n\n

Example : <\/span>Evaluate : \\(\\displaystyle{\\lim_{x \\to 1}}\\) \\({(log_3 3x)}^{log_x 3}\\) <\/p>\n

Solution : <\/span>\n \\(\\displaystyle{\\lim_{x \\to 1}}\\) \\({(log_3 3x)}^{log_x 3}\\) = \\(\\displaystyle{\\lim_{x \\to 1}}\\)\n \\({(log_3 3 + log_3 x)}^{log_x 3}\\)

= \\(\\displaystyle{\\lim_{x \\to 1}}\\) \\({(1 + log_3 x)}^{1\/log_3 x}\\) = e

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

\n
Previous – How to find limit of trigonometric functions <\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn what is squeeze theorem or sandwich theorem of limit and limit of exponential function with examples. Let’s begin – Squeeze Theorem (Sandwich Theorem) If f(x) \\(\\leq\\) g(x) \\(\\leq\\) h(x); \\(\\forall\\) x in the neighbourhood at x = a and \\(\\displaystyle{\\lim_{x \\to a}}\\) f(x) = l = \\(\\displaystyle{\\lim_{x \\to 1}}\\) h(x) then …<\/p>\n

What is Squeeze Theorem – Limit of Exponential 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":[450,449,448],"yoast_head":"\nWhat is Squeeze Theorem - Limit of Exponential Functions<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is squeeze theorem or sandwich theorem of limit and limit of exponential function with examples.\" \/>\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\/what-is-squeeze-theorem\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is Squeeze Theorem - Limit of Exponential Functions\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn what is squeeze theorem or sandwich theorem of limit and limit of exponential function with examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-08-11T13:12:22+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-26T11:05:56+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\/what-is-squeeze-theorem\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"What is Squeeze Theorem – Limit of Exponential Functions\",\"datePublished\":\"2021-08-11T13:12:22+00:00\",\"dateModified\":\"2021-11-26T11:05:56+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/\"},\"wordCount\":363,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"limit of exponential functions\",\"sandwich theorem\",\"squeeze theorem limits\"],\"articleSection\":[\"Limits\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/\",\"url\":\"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/\",\"name\":\"What is Squeeze Theorem - Limit of Exponential Functions\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-08-11T13:12:22+00:00\",\"dateModified\":\"2021-11-26T11:05:56+00:00\",\"description\":\"In this post you will learn what is squeeze theorem or sandwich theorem of limit and limit of exponential function with examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"What is Squeeze Theorem – Limit of Exponential 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":"What is Squeeze Theorem - Limit of Exponential Functions","description":"In this post you will learn what is squeeze theorem or sandwich theorem of limit and limit of exponential function with examples.","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\/what-is-squeeze-theorem\/","og_locale":"en_US","og_type":"article","og_title":"What is Squeeze Theorem - Limit of Exponential Functions","og_description":"In this post you will learn what is squeeze theorem or sandwich theorem of limit and limit of exponential function with examples.","og_url":"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/","og_site_name":"Mathemerize","article_published_time":"2021-08-11T13:12:22+00:00","article_modified_time":"2021-11-26T11:05:56+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\/what-is-squeeze-theorem\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"What is Squeeze Theorem – Limit of Exponential Functions","datePublished":"2021-08-11T13:12:22+00:00","dateModified":"2021-11-26T11:05:56+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/"},"wordCount":363,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"keywords":["limit of exponential functions","sandwich theorem","squeeze theorem limits"],"articleSection":["Limits"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/what-is-squeeze-theorem\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/","url":"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/","name":"What is Squeeze Theorem - Limit of Exponential Functions","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-08-11T13:12:22+00:00","dateModified":"2021-11-26T11:05:56+00:00","description":"In this post you will learn what is squeeze theorem or sandwich theorem of limit and limit of exponential function with examples.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/what-is-squeeze-theorem\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/what-is-squeeze-theorem\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"What is Squeeze Theorem – Limit of Exponential 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\/3961"}],"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=3961"}],"version-history":[{"count":7,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/3961\/revisions"}],"predecessor-version":[{"id":5325,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/3961\/revisions\/5325"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=3961"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=3961"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=3961"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}