{"id":319,"date":"2021-02-04T01:09:54","date_gmt":"2021-02-04T01:09:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=319"},"modified":"2021-03-18T15:20:37","modified_gmt":"2021-03-18T15:20:37","slug":"summary-of-the-limit-of-a-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-the-limit-of-a-function\/","title":{"raw":"Summary of the Limit of a Function","rendered":"Summary of the Limit of a Function"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>A table of values or graph may be used to estimate a limit.<\/li>\r\n \t<li>If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist.<\/li>\r\n \t<li>If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value.<\/li>\r\n \t<li>We may use limits to describe infinite behavior of a function at a point.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572347681\">\r\n \t<li><strong>One-Sided Limits<\/strong>\r\n[latex]\\underset{x\\to a^-}{\\lim}f(x)=L[\/latex]\r\n[latex]\\underset{x\\to a^+}{\\lim}f(x)=L[\/latex]<\/li>\r\n \t<li><strong>Intuitive Definition of the Limit<\/strong>\r\n[latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572541944\" class=\"definition\">\r\n \t<dt>infinite limit<\/dt>\r\n \t<dd id=\"fs-id1170572541950\">A function has an infinite limit at a point [latex]a[\/latex] if it either increases or decreases without bound as it approaches [latex]a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572467930\" class=\"definition\">\r\n \t<dt>intuitive definition of the limit<\/dt>\r\n \t<dd id=\"fs-id1170572467935\">If all values of the function [latex]f(x)[\/latex] approach the real number [latex]L[\/latex] as the values of [latex]x(\\ne a)[\/latex] approach [latex]a[\/latex], [latex]f(x)[\/latex] approaches [latex]L[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572467997\" class=\"definition\">\r\n \t<dt>one-sided limit<\/dt>\r\n \t<dd id=\"fs-id1170572468002\">A one-sided limit of a function is a limit taken from either the left or the right<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572468006\" class=\"definition\">\r\n \t<dt>vertical asymptote<\/dt>\r\n \t<dd id=\"fs-id1170572468012\">A function has a vertical asymptote at [latex]x=a[\/latex] if the limit as [latex]x[\/latex] approaches [latex]a[\/latex] from the right or left is infinite<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>A table of values or graph may be used to estimate a limit.<\/li>\n<li>If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist.<\/li>\n<li>If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value.<\/li>\n<li>We may use limits to describe infinite behavior of a function at a point.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572347681\">\n<li><strong>One-Sided Limits<\/strong><br \/>\n[latex]\\underset{x\\to a^-}{\\lim}f(x)=L[\/latex]<br \/>\n[latex]\\underset{x\\to a^+}{\\lim}f(x)=L[\/latex]<\/li>\n<li><strong>Intuitive Definition of the Limit<\/strong><br \/>\n[latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572541944\" class=\"definition\">\n<dt>infinite limit<\/dt>\n<dd id=\"fs-id1170572541950\">A function has an infinite limit at a point [latex]a[\/latex] if it either increases or decreases without bound as it approaches [latex]a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572467930\" class=\"definition\">\n<dt>intuitive definition of the limit<\/dt>\n<dd id=\"fs-id1170572467935\">If all values of the function [latex]f(x)[\/latex] approach the real number [latex]L[\/latex] as the values of [latex]x(\\ne a)[\/latex] approach [latex]a[\/latex], [latex]f(x)[\/latex] approaches [latex]L[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572467997\" class=\"definition\">\n<dt>one-sided limit<\/dt>\n<dd id=\"fs-id1170572468002\">A one-sided limit of a function is a limit taken from either the left or the right<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572468006\" class=\"definition\">\n<dt>vertical asymptote<\/dt>\n<dd id=\"fs-id1170572468012\">A function has a vertical asymptote at [latex]x=a[\/latex] if the limit as [latex]x[\/latex] approaches [latex]a[\/latex] from the right or left is infinite<\/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-319\">\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":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 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https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-319","chapter","type-chapter","status-publish","hentry"],"part":28,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/319","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/319\/revisions"}],"predecessor-version":[{"id":1428,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/319\/revisions\/1428"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/28"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/319\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=319"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=319"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=319"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=319"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}