{"id":322,"date":"2021-02-04T01:10:27","date_gmt":"2021-02-04T01:10:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=322"},"modified":"2021-03-23T01:01:46","modified_gmt":"2021-03-23T01:01:46","slug":"summary-of-the-precise-definition-of-a-limit","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-the-precise-definition-of-a-limit\/","title":{"raw":"Summary of the Precise Definition of a Limit","rendered":"Summary of the Precise Definition of a Limit"},"content":{"raw":"<div id=\"fs-id1170572551843\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170572551850\">\r\n \t<li>The intuitive notion of a limit may be converted into a rigorous mathematical definition known as the epsilon-delta definition of the limit.<\/li>\r\n \t<li>The epsilon-delta definition may be used to prove statements about limits.<\/li>\r\n \t<li>The epsilon-delta definition of a limit may be used to find deltas algebraically.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<dl id=\"fs-id1170572386184\" class=\"definition\">\r\n \t<dt>\r\n<h2><strong>Glossary<\/strong><\/h2>\r\n<\/dt>\r\n \t<dt><\/dt>\r\n \t<dt>epsilon-delta definition of the limit<\/dt>\r\n \t<dd id=\"fs-id1170572386190\">[latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex] if for every [latex]\\varepsilon &gt;0[\/latex], there exists a [latex]\\delta &gt;0[\/latex] such that if [latex]0&lt;|x-a|&lt;\\delta[\/latex], then [latex]|f(x)-L|&lt;\\varepsilon [\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572643277\" class=\"definition\">\r\n \t<dt>triangle inequality<\/dt>\r\n \t<dd id=\"fs-id1170572373658\">If [latex]a[\/latex] and [latex]b[\/latex] are any real numbers, then [latex]|a+b|\\le |a|+|b|[\/latex]<\/dd>\r\n<\/dl>\r\n&nbsp;\r\n<dl id=\"fs-id1170572643277\" class=\"definition\">\r\n \t<dd><\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1170572551843\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170572551850\">\n<li>The intuitive notion of a limit may be converted into a rigorous mathematical definition known as the epsilon-delta definition of the limit.<\/li>\n<li>The epsilon-delta definition may be used to prove statements about limits.<\/li>\n<li>The epsilon-delta definition of a limit may be used to find deltas algebraically.<\/li>\n<\/ul>\n<\/div>\n<dl id=\"fs-id1170572386184\" class=\"definition\">\n<dt>\n<\/dt>\n<dt><\/dt>\n<dt>epsilon-delta definition of the limit<\/dt>\n<dd id=\"fs-id1170572386190\">[latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex] if for every [latex]\\varepsilon >0[\/latex], there exists a [latex]\\delta >0[\/latex] such that if [latex]0<|x-a|<\\delta[\/latex], then [latex]|f(x)-L|<\\varepsilon[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572643277\" class=\"definition\">\n<dt>triangle inequality<\/dt>\n<dd id=\"fs-id1170572373658\">If [latex]a[\/latex] and [latex]b[\/latex] are any real numbers, then [latex]|a+b|\\le |a|+|b|[\/latex]<\/dd>\n<\/dl>\n<p>&nbsp;<\/p>\n<dl id=\"fs-id1170572643277\" class=\"definition\">\n<dd><\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-322\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":23,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at 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