{"id":2163,"date":"2021-03-27T18:16:08","date_gmt":"2021-03-27T18:16:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=2163"},"modified":"2021-04-03T02:19:37","modified_gmt":"2021-04-03T02:19:37","slug":"summary-of-limits-at-infinity-and-asymptotes","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-limits-at-infinity-and-asymptotes\/","title":{"raw":"Summary of Limits at Infinity and Asymptotes","rendered":"Summary of Limits at Infinity and Asymptotes"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>If [latex]c[\/latex] is a critical point of [latex]f[\/latex] and [latex]f^{\\prime}(x)&gt;0[\/latex] for [latex]x&lt;c[\/latex] and [latex]f^{\\prime}(x)&lt;0[\/latex] for [latex]x&gt;c[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\r\n \t<li>If [latex]c[\/latex] is a critical point of [latex]f[\/latex] and [latex]f^{\\prime}(x)&lt;0[\/latex] for [latex]x&lt;c[\/latex] and [latex]f^{\\prime}(x)&gt;0[\/latex] for [latex]x&gt;c[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\r\n \t<li>For a polynomial function [latex]p(x)=a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0[\/latex], where [latex]a_n \\ne 0[\/latex], the end behavior is determined by the leading term [latex]a_n x^n[\/latex]. If [latex]n\\ne 0[\/latex], [latex]p(x)[\/latex] approaches [latex]\\infty [\/latex] or [latex]\u2212\\infty [\/latex] at each end.<\/li>\r\n \t<li>For a rational function [latex]f(x)=\\frac{p(x)}{q(x)}[\/latex], the end behavior is determined by the relationship between the degree of [latex]p[\/latex] and the degree of [latex]q[\/latex]. If the degree of [latex]p[\/latex] is less than the degree of [latex]q[\/latex], the line [latex]y=0[\/latex] is a horizontal asymptote for [latex]f[\/latex]. If the degree of [latex]p[\/latex] is equal to the degree of [latex]q[\/latex], then the line [latex]y=\\frac{a_n}{b_n}[\/latex] is a horizontal asymptote, where [latex]a_n[\/latex] and [latex]b_n[\/latex] are the leading coefficients of [latex]p[\/latex] and [latex]q[\/latex], respectively. If the degree of [latex]p[\/latex] is greater than the degree of [latex]q[\/latex], then [latex]f[\/latex] approaches [latex]\\infty [\/latex] or [latex]\u2212\\infty [\/latex] at each end.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572347681\">\r\n \t<li><strong>Infinite Limits from the Left<\/strong>\r\n[latex]\\underset{x\\to a^-}{\\lim}f(x)=+\\infty[\/latex]\r\n[latex]\\underset{x\\to a^-}{\\lim}f(x)=\u2212\\infty [\/latex]<\/li>\r\n \t<li><strong>Infinite Limits from the Right<\/strong>\r\n[latex]\\underset{x\\to a^+}{\\lim}f(x)=+\\infty[\/latex]\r\n[latex]\\underset{x\\to a^+}{\\lim}f(x)=\u2212\\infty [\/latex]<\/li>\r\n \t<li><strong>Two-Sided Infinite Limits<\/strong>\r\n[latex]\\underset{x\\to a}{\\lim}f(x)=+\\infty: \\underset{x\\to a^-}{\\lim}f(x)=+\\infty[\/latex] and [latex]\\underset{x\\to a^+}{\\lim}f(x)=+\\infty[\/latex]\r\n[latex]\\underset{x\\to a}{\\lim}f(x)=\u2212\\infty: \\underset{x\\to a^-}{\\lim}f(x)=\u2212\\infty [\/latex] and [latex]\\underset{x\\to a^+}{\\lim}f(x)=\u2212\\infty [\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165043208865\" class=\"definition\">\r\n \t<dt>end behavior<\/dt>\r\n \t<dd id=\"fs-id1165043208870\">the behavior of a function as [latex]x\\to \\infty [\/latex] and [latex]x\\to \u2212\\infty [\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043208899\" class=\"definition\">\r\n \t<dt>horizontal asymptote<\/dt>\r\n \t<dd id=\"fs-id1165043208905\">if [latex]\\underset{x\\to \\infty }{\\lim}f(x)=L[\/latex] or [latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)=L[\/latex], then [latex]y=L[\/latex] is a horizontal asymptote of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042462524\" class=\"definition\">\r\n \t<dt>infinite limit at infinity<\/dt>\r\n \t<dd id=\"fs-id1165042462530\">a function that becomes arbitrarily large as [latex]x[\/latex] becomes large<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042462539\" class=\"definition\">\r\n \t<dt>limit at infinity<\/dt>\r\n \t<dd id=\"fs-id1165042462545\">the limiting value, if it exists, of a function as [latex]x\\to \\infty [\/latex] or [latex]x\\to \u2212\\infty [\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042462574\" class=\"definition\">\r\n \t<dt>oblique asymptote<\/dt>\r\n \t<dd id=\"fs-id1165042462579\">the line [latex]y=mx+b[\/latex] if [latex]f(x)[\/latex] approaches it as [latex]x\\to \\infty [\/latex] or [latex]x\\to \u2212\\infty [\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>If [latex]c[\/latex] is a critical point of [latex]f[\/latex] and [latex]f^{\\prime}(x)>0[\/latex] for [latex]x<c[\/latex] and [latex]f^{\\prime}(x)<0[\/latex] for [latex]x>c[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\n<li>If [latex]c[\/latex] is a critical point of [latex]f[\/latex] and [latex]f^{\\prime}(x)<0[\/latex] for [latex]x<c[\/latex] and [latex]f^{\\prime}(x)>0[\/latex] for [latex]x>c[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\n<li>For a polynomial function [latex]p(x)=a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0[\/latex], where [latex]a_n \\ne 0[\/latex], the end behavior is determined by the leading term [latex]a_n x^n[\/latex]. If [latex]n\\ne 0[\/latex], [latex]p(x)[\/latex] approaches [latex]\\infty[\/latex] or [latex]\u2212\\infty[\/latex] at each end.<\/li>\n<li>For a rational function [latex]f(x)=\\frac{p(x)}{q(x)}[\/latex], the end behavior is determined by the relationship between the degree of [latex]p[\/latex] and the degree of [latex]q[\/latex]. If the degree of [latex]p[\/latex] is less than the degree of [latex]q[\/latex], the line [latex]y=0[\/latex] is a horizontal asymptote for [latex]f[\/latex]. If the degree of [latex]p[\/latex] is equal to the degree of [latex]q[\/latex], then the line [latex]y=\\frac{a_n}{b_n}[\/latex] is a horizontal asymptote, where [latex]a_n[\/latex] and [latex]b_n[\/latex] are the leading coefficients of [latex]p[\/latex] and [latex]q[\/latex], respectively. If the degree of [latex]p[\/latex] is greater than the degree of [latex]q[\/latex], then [latex]f[\/latex] approaches [latex]\\infty[\/latex] or [latex]\u2212\\infty[\/latex] at each end.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572347681\">\n<li><strong>Infinite Limits from the Left<\/strong><br \/>\n[latex]\\underset{x\\to a^-}{\\lim}f(x)=+\\infty[\/latex]<br \/>\n[latex]\\underset{x\\to a^-}{\\lim}f(x)=\u2212\\infty[\/latex]<\/li>\n<li><strong>Infinite Limits from the Right<\/strong><br \/>\n[latex]\\underset{x\\to a^+}{\\lim}f(x)=+\\infty[\/latex]<br \/>\n[latex]\\underset{x\\to a^+}{\\lim}f(x)=\u2212\\infty[\/latex]<\/li>\n<li><strong>Two-Sided Infinite Limits<\/strong><br \/>\n[latex]\\underset{x\\to a}{\\lim}f(x)=+\\infty: \\underset{x\\to a^-}{\\lim}f(x)=+\\infty[\/latex] and [latex]\\underset{x\\to a^+}{\\lim}f(x)=+\\infty[\/latex]<br \/>\n[latex]\\underset{x\\to a}{\\lim}f(x)=\u2212\\infty: \\underset{x\\to a^-}{\\lim}f(x)=\u2212\\infty[\/latex] and [latex]\\underset{x\\to a^+}{\\lim}f(x)=\u2212\\infty[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165043208865\" class=\"definition\">\n<dt>end behavior<\/dt>\n<dd id=\"fs-id1165043208870\">the behavior of a function as [latex]x\\to \\infty[\/latex] and [latex]x\\to \u2212\\infty[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043208899\" class=\"definition\">\n<dt>horizontal asymptote<\/dt>\n<dd id=\"fs-id1165043208905\">if [latex]\\underset{x\\to \\infty }{\\lim}f(x)=L[\/latex] or [latex]\\underset{x\\to \u2212\\infty }{\\lim}f(x)=L[\/latex], then [latex]y=L[\/latex] is a horizontal asymptote of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042462524\" class=\"definition\">\n<dt>infinite limit at infinity<\/dt>\n<dd id=\"fs-id1165042462530\">a function that becomes arbitrarily large as [latex]x[\/latex] becomes large<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042462539\" class=\"definition\">\n<dt>limit at infinity<\/dt>\n<dd id=\"fs-id1165042462545\">the limiting value, if it exists, of a function as [latex]x\\to \\infty[\/latex] or [latex]x\\to \u2212\\infty[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042462574\" class=\"definition\">\n<dt>oblique asymptote<\/dt>\n<dd id=\"fs-id1165042462579\">the line [latex]y=mx+b[\/latex] if [latex]f(x)[\/latex] approaches it as [latex]x\\to \\infty[\/latex] or [latex]x\\to \u2212\\infty[\/latex]<\/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-2163\">\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 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-2163","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2163","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":3,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2163\/revisions"}],"predecessor-version":[{"id":2625,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2163\/revisions\/2625"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/48"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2163\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=2163"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=2163"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=2163"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=2163"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}