{"id":3589,"date":"2024-06-28T17:11:00","date_gmt":"2024-06-28T17:11:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3589"},"modified":"2024-08-04T11:06:39","modified_gmt":"2024-08-04T11:06:39","slug":"understanding-limits-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/understanding-limits-cheat-sheet\/","title":{"raw":"Understanding Limits: Cheat Sheet","rendered":"Understanding Limits: Cheat Sheet"},"content":{"raw":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Understanding+Limits.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\r\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\r\n<h2>Essential Concepts<\/h2>\r\n<p><strong>A Preview of Calculus<\/strong><\/p>\r\n<ul>\r\n\t<li>Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the\u00a0<em data-effect=\"italics\">derivative<\/em>. Calculating a derivative requires finding a limit.<\/li>\r\n\t<li>Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the [latex]x[\/latex]-axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called the\u00a0<em data-effect=\"italics\">integral<\/em>. The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function.<\/li>\r\n\t<li>Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in space and finding volumes of solids.<\/li>\r\n<\/ul>\r\n<p><strong>Introduction to the limit of a function<\/strong><\/p>\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<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572347681\">\r\n\t<li><strong>Slope of a Secant Line<\/strong><br \/>\r\n[latex]m_{\\sec}=\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/li>\r\n\t<li><strong>Average Velocity over Interval\u00a0<\/strong>[latex][a,t][\/latex]<br \/>\r\n[latex]v_{\\text{avg}}=\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/li>\r\n\t<li><strong>One-Sided Limits<\/strong><br \/>\r\n[latex]\\underset{x\\to a^-}{\\lim}f(x)=L[\/latex]<br \/>\r\n[latex]\\underset{x\\to a^+}{\\lim}f(x)=L[\/latex]<\/li>\r\n\t<li><strong>Intuitive Definition of the Limit<\/strong><br \/>\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<dt>infinite limit<\/dt>\r\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>\r\n<\/dl>\r\n<dl id=\"fs-id1170572467930\" class=\"definition\">\r\n<dt>intuitive definition of the limit<\/dt>\r\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>\r\n<\/dl>\r\n<dl id=\"fs-id1170572467997\" class=\"definition\">\r\n<dt>one-sided limit<\/dt>\r\n<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<dt>vertical asymptote<\/dt>\r\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>\r\n<\/dl>\r\n<h2>Study Tips<\/h2>\r\n<p><strong>The Tangent Problem and Differential Calculus<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice sketching secant lines for various functions and intervals.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize how secant lines approach the tangent line as points get closer.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When working with position functions, relate concepts of average and instantaneous velocity to secant and tangent lines.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice calculating average velocities over smaller and smaller intervals.<\/li>\r\n<\/ul>\r\n<p><strong>The Area Problem and Integral Calculus<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice approximating areas using different numbers of rectangles. Understand the relationship between smaller rectangles and better approximations.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize how the sum of rectangle areas approaches the true area as widths decrease.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice setting up integrals for various shaped regions.<\/li>\r\n<\/ul>\r\n<p><strong>The Definition of a Limit<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice estimating limits using both tables and graphs.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Be aware that a function can be defined at a point but have a different limit there.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Look for oscillating behavior when suspecting a limit might not exist.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that limits describe behavior near a point, not necessarily at the point.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use basic limit properties to simplify complex limit problems.<\/li>\r\n<\/ul>\r\n<p><strong>One-Sided and Two-Sided Limits<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">When evaluating limits, always consider both left-hand and right-hand approaches.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For piecewise functions, use the appropriate piece for each one-sided limit.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Create tables of values approaching the point of interest from both sides.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that a function can be continuous from one side but not the other.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When solving limit problems, explicitly state both one-sided limits before concluding about the two-sided limit.<\/li>\r\n<\/ul>\r\n<p><strong>Infinite Limits<\/strong><\/p>\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Remember that infinite limits are not actual limits but descriptions of behavior.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When finding vertical asymptotes, check both sides of the potential asymptote.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use factoring to identify potential vertical asymptotes in rational functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Distinguish between vertical asymptotes and holes in rational functions.<\/li>\r\n<\/ul>","rendered":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Understanding+Limits.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<p><strong>A Preview of Calculus<\/strong><\/p>\n<ul>\n<li>Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the\u00a0<em data-effect=\"italics\">derivative<\/em>. Calculating a derivative requires finding a limit.<\/li>\n<li>Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the [latex]x[\/latex]-axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called the\u00a0<em data-effect=\"italics\">integral<\/em>. The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function.<\/li>\n<li>Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in space and finding volumes of solids.<\/li>\n<\/ul>\n<p><strong>Introduction to the limit of a function<\/strong><\/p>\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<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572347681\">\n<li><strong>Slope of a Secant Line<\/strong><br \/>\n[latex]m_{\\sec}=\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/li>\n<li><strong>Average Velocity over Interval\u00a0<\/strong>[latex][a,t][\/latex]<br \/>\n[latex]v_{\\text{avg}}=\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/li>\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<h2>Study Tips<\/h2>\n<p><strong>The Tangent Problem and Differential Calculus<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice sketching secant lines for various functions and intervals.<\/li>\n<li class=\"whitespace-normal break-words\">Visualize how secant lines approach the tangent line as points get closer.<\/li>\n<li class=\"whitespace-normal break-words\">When working with position functions, relate concepts of average and instantaneous velocity to secant and tangent lines.<\/li>\n<li class=\"whitespace-normal break-words\">Practice calculating average velocities over smaller and smaller intervals.<\/li>\n<\/ul>\n<p><strong>The Area Problem and Integral Calculus<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice approximating areas using different numbers of rectangles. Understand the relationship between smaller rectangles and better approximations.<\/li>\n<li class=\"whitespace-normal break-words\">Visualize how the sum of rectangle areas approaches the true area as widths decrease.<\/li>\n<li class=\"whitespace-normal break-words\">Practice setting up integrals for various shaped regions.<\/li>\n<\/ul>\n<p><strong>The Definition of a Limit<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice estimating limits using both tables and graphs.<\/li>\n<li class=\"whitespace-normal break-words\">Be aware that a function can be defined at a point but have a different limit there.<\/li>\n<li class=\"whitespace-normal break-words\">Look for oscillating behavior when suspecting a limit might not exist.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that limits describe behavior near a point, not necessarily at the point.<\/li>\n<li class=\"whitespace-normal break-words\">Use basic limit properties to simplify complex limit problems.<\/li>\n<\/ul>\n<p><strong>One-Sided and Two-Sided Limits<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">When evaluating limits, always consider both left-hand and right-hand approaches.<\/li>\n<li class=\"whitespace-normal break-words\">For piecewise functions, use the appropriate piece for each one-sided limit.<\/li>\n<li class=\"whitespace-normal break-words\">Create tables of values approaching the point of interest from both sides.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that a function can be continuous from one side but not the other.<\/li>\n<li class=\"whitespace-normal break-words\">When solving limit problems, explicitly state both one-sided limits before concluding about the two-sided limit.<\/li>\n<\/ul>\n<p><strong>Infinite Limits<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Remember that infinite limits are not actual limits but descriptions of behavior.<\/li>\n<li class=\"whitespace-normal break-words\">When finding vertical asymptotes, check both sides of the potential asymptote.<\/li>\n<li class=\"whitespace-normal break-words\">Use factoring to identify potential vertical asymptotes in rational functions.<\/li>\n<li class=\"whitespace-normal break-words\">Distinguish between vertical asymptotes and holes in rational functions.<\/li>\n<\/ul>\n","protected":false},"author":15,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":484,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3589"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3589\/revisions"}],"predecessor-version":[{"id":3948,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3589\/revisions\/3948"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/484"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3589\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3589"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3589"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3589"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3589"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}