{"id":320,"date":"2021-02-04T01:10:03","date_gmt":"2021-02-04T01:10:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=320"},"modified":"2021-03-25T00:06:36","modified_gmt":"2021-03-25T00:06:36","slug":"summary-of-the-limit-laws","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-the-limit-laws\/","title":{"raw":"Summary of the Limit Laws","rendered":"Summary of the Limit Laws"},"content":{"raw":"<div id=\"fs-id1170572624436\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170572624443\">\r\n \t<li>The limit laws allow us to evaluate limits of functions without having to go through step-by-step processes each time.<\/li>\r\n \t<li>For polynomials and rational functions, [latex]\\underset{x\\to a}{\\lim}f(x)=f(a)[\/latex].<\/li>\r\n \t<li>You can evaluate the limit of a function by factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction.<\/li>\r\n \t<li>The Squeeze Theorem allows you to find the limit of a function if the function is always greater than one function and less than another function with limits that are known.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572347681\">\r\n \t<li><strong>Basic Limit Results<\/strong>\r\n[latex]\\underset{x\\to a}{\\lim}x=a[\/latex]\r\n[latex]\\underset{x\\to a}{\\lim}c=c[\/latex]<\/li>\r\n \t<li><strong>Important Limits<\/strong>\r\n[latex]\\underset{\\theta \\to 0}{\\lim} \\sin \\theta =0[\/latex]\r\n[latex]\\underset{\\theta \\to 0}{\\lim} \\cos \\theta =1[\/latex]\r\n[latex]\\underset{\\theta \\to 0}{\\lim}\\frac{\\sin \\theta}{\\theta}=1[\/latex]\r\n[latex]\\underset{\\theta \\to 0}{\\lim}\\frac{1- \\cos \\theta}{\\theta}=0[\/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>\r\n&nbsp;","rendered":"<div id=\"fs-id1170572624436\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170572624443\">\n<li>The limit laws allow us to evaluate limits of functions without having to go through step-by-step processes each time.<\/li>\n<li>For polynomials and rational functions, [latex]\\underset{x\\to a}{\\lim}f(x)=f(a)[\/latex].<\/li>\n<li>You can evaluate the limit of a function by factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction.<\/li>\n<li>The Squeeze Theorem allows you to find the limit of a function if the function is always greater than one function and less than another function with limits that are known.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572347681\">\n<li><strong>Basic Limit Results<\/strong><br \/>\n[latex]\\underset{x\\to a}{\\lim}x=a[\/latex]<br \/>\n[latex]\\underset{x\\to a}{\\lim}c=c[\/latex]<\/li>\n<li><strong>Important Limits<\/strong><br \/>\n[latex]\\underset{\\theta \\to 0}{\\lim} \\sin \\theta =0[\/latex]<br \/>\n[latex]\\underset{\\theta \\to 0}{\\lim} \\cos \\theta =1[\/latex]<br \/>\n[latex]\\underset{\\theta \\to 0}{\\lim}\\frac{\\sin \\theta}{\\theta}=1[\/latex]<br \/>\n[latex]\\underset{\\theta \\to 0}{\\lim}\\frac{1- \\cos \\theta}{\\theta}=0[\/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<p>&nbsp;<\/p>\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-320\">\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":15,"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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