{"id":1634,"date":"2021-03-19T20:16:50","date_gmt":"2021-03-19T20:16:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1634"},"modified":"2021-04-30T18:28:41","modified_gmt":"2021-04-30T18:28:41","slug":"summary-of-lhopitals-rule","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-lhopitals-rule\/","title":{"raw":"Summary of L\u2019H\u00f4pital\u2019s Rule","rendered":"Summary of L\u2019H\u00f4pital\u2019s Rule"},"content":{"raw":"<div id=\"fs-id1165042658525\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1165042658532\">\r\n \t<li>L\u2019H\u00f4pital\u2019s rule can be used to evaluate the limit of a quotient when the indeterminate form [latex]\\frac{0}{0}[\/latex] or [latex]\\frac{\\infty}{\\infty}[\/latex] arises.<\/li>\r\n \t<li>L\u2019H\u00f4pital\u2019s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form [latex]\\frac{0}{0}[\/latex] or [latex]\\frac{\\infty}{\\infty}[\/latex].<\/li>\r\n \t<li>The exponential function [latex]e^x[\/latex] grows faster than any power function [latex]x^p[\/latex], [latex]p&gt;0[\/latex].<\/li>\r\n \t<li>The logarithmic function [latex]\\ln x[\/latex] grows more slowly than any power function [latex]x^p[\/latex], [latex]p&gt;0[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165042539157\" class=\"definition\">\r\n \t<dt>indeterminate forms<\/dt>\r\n \t<dd id=\"fs-id1165042539162\">when evaluating a limit, the forms [latex]0\/0[\/latex], [latex]\\infty \/ \\infty[\/latex], [latex]0 \\cdot \\infty[\/latex], [latex]\\infty -\\infty[\/latex], [latex]0^0[\/latex], [latex]\\infty^0[\/latex], and [latex]1^{\\infty}[\/latex] are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042539243\" class=\"definition\">\r\n \t<dt>L\u2019H\u00f4pital\u2019s rule<\/dt>\r\n \t<dd id=\"fs-id1165042539249\">if [latex]f[\/latex] and [latex]g[\/latex] are differentiable functions over an interval [latex]a[\/latex], except possibly at [latex]a[\/latex], and [latex]\\underset{x\\to a}{\\lim} f(x)=0=\\underset{x\\to a}{\\lim} g(x)[\/latex] or [latex]\\underset{x\\to a}{\\lim} f(x)[\/latex] and [latex]\\underset{x\\to a}{\\lim} g(x)[\/latex] are infinite, then [latex]\\underset{x\\to a}{\\lim}\\dfrac{f(x)}{g(x)}=\\underset{x\\to a}{\\lim}\\dfrac{f^{\\prime}(x)}{g^{\\prime}(x)}[\/latex], assuming the limit on the right exists or is [latex]\\infty [\/latex] or [latex]\u2212\\infty [\/latex]<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1165042658525\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1165042658532\">\n<li>L\u2019H\u00f4pital\u2019s rule can be used to evaluate the limit of a quotient when the indeterminate form [latex]\\frac{0}{0}[\/latex] or [latex]\\frac{\\infty}{\\infty}[\/latex] arises.<\/li>\n<li>L\u2019H\u00f4pital\u2019s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form [latex]\\frac{0}{0}[\/latex] or [latex]\\frac{\\infty}{\\infty}[\/latex].<\/li>\n<li>The exponential function [latex]e^x[\/latex] grows faster than any power function [latex]x^p[\/latex], [latex]p>0[\/latex].<\/li>\n<li>The logarithmic function [latex]\\ln x[\/latex] grows more slowly than any power function [latex]x^p[\/latex], [latex]p>0[\/latex].<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165042539157\" class=\"definition\">\n<dt>indeterminate forms<\/dt>\n<dd id=\"fs-id1165042539162\">when evaluating a limit, the forms [latex]0\/0[\/latex], [latex]\\infty \/ \\infty[\/latex], [latex]0 \\cdot \\infty[\/latex], [latex]\\infty -\\infty[\/latex], [latex]0^0[\/latex], [latex]\\infty^0[\/latex], and [latex]1^{\\infty}[\/latex] are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042539243\" class=\"definition\">\n<dt>L\u2019H\u00f4pital\u2019s rule<\/dt>\n<dd id=\"fs-id1165042539249\">if [latex]f[\/latex] and [latex]g[\/latex] are differentiable functions over an interval [latex]a[\/latex], except possibly at [latex]a[\/latex], and [latex]\\underset{x\\to a}{\\lim} f(x)=0=\\underset{x\\to a}{\\lim} g(x)[\/latex] or [latex]\\underset{x\\to a}{\\lim} f(x)[\/latex] and [latex]\\underset{x\\to a}{\\lim} g(x)[\/latex] are infinite, then [latex]\\underset{x\\to a}{\\lim}\\dfrac{f(x)}{g(x)}=\\underset{x\\to a}{\\lim}\\dfrac{f^{\\prime}(x)}{g^{\\prime}(x)}[\/latex], assuming the limit on the right exists or is [latex]\\infty[\/latex] or [latex]\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-1634\">\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":31,"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-1634","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1634","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\/1634\/revisions"}],"predecessor-version":[{"id":3507,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1634\/revisions\/3507"}],"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\/1634\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1634"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1634"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1634"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1634"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}