{"id":264,"date":"2025-02-13T22:45:45","date_gmt":"2025-02-13T22:45:45","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/finding-limits-numerical-and-graphical-approaches\/"},"modified":"2025-10-23T19:16:53","modified_gmt":"2025-10-23T19:16:53","slug":"finding-limits-numerical-and-graphical-approaches","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/finding-limits-numerical-and-graphical-approaches\/","title":{"raw":"Finding Limits: Learn It 1","rendered":"Finding Limits: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Find a limit using a graph.<\/li>\r\n \t<li style=\"font-weight: 400;\">Find a limit using a table.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Understanding Limit Notation<\/h2>\r\nWe have seen how a <strong>sequence<\/strong> can have a limit, a value that the sequence of terms moves toward as the nu mber of terms increases. For example, the terms of the sequence\r\n<div style=\"text-align: center;\">[latex]1,\\frac{1}{2},\\frac{1}{4},\\frac{1}{8}..[\/latex].<\/div>\r\ngets closer and closer to 0. A sequence is one type of function, but functions that are not sequences can also have limits. We can describe the behavior of the function as the input values get close to a specific value. If the limit of a function [latex]f\\left(x\\right)=L[\/latex], then as the input [latex]x[\/latex] gets closer and closer to [latex]a[\/latex], the output <em>y<\/em>-coordinate gets closer and closer to [latex]L[\/latex]. We say that the output \"approaches\" [latex]L[\/latex].\r\n\r\nAs the input value [latex]x[\/latex] approaches [latex]a[\/latex], the output value [latex]f\\left(x\\right)[\/latex] approaches [latex]L[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185219\/CNX_Precalc_Figure_12_01_0012.jpg\" alt=\"Graph representing how a function with a hole at (a, L) approaches a limit.\" width=\"487\" height=\"405\" \/> The output (y-coordinate) approaches [latex]L[\/latex] as the input (x-coordinate) approaches [latex]a[\/latex].[\/caption]We write the equation of a limit as\r\n<div style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/div>\r\nThis notation indicates that as [latex]x[\/latex] approaches [latex]a[\/latex] both from the left of [latex]x=a[\/latex] and the right of [latex]x=a[\/latex], the output value approaches [latex]L[\/latex].\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Consider the function\r\n<div style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{{x}^{2}-6x - 7}{x - 7}[\/latex].<\/div>\r\nWe can factor the function as shown.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;f\\left(x\\right)=\\frac{\\cancel{\\left(x - 7\\right)}\\left(x+1\\right)}{\\cancel{x - 7}}&amp;&amp; \\text{Cancel like factors in numerator and denominator.} \\\\ &amp;f\\left(x\\right)=x+1,x\\ne 7&amp;&amp; \\text{Simplify}. \\end{align}[\/latex]<\/div>\r\nNotice that [latex]x[\/latex] cannot be 7, or we would be dividing by 0, so 7 is not in the domain of the original function. In order to avoid changing the function when we simplify, we set the same condition, [latex]x\\ne 7[\/latex], for the simplified function. We can represent the function graphically.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185222\/CNX_Precalc_Figure_12_01_0022.jpg\" alt=\"Graph of an increasing function, f(x) = (x^2-6x-7)\/(x-7), with a hole at (7, 8).\" width=\"487\" height=\"483\" \/> Because 7 is not allowed as an input, there is no point at [latex]x=7[\/latex].[\/caption]What happens at [latex]x=7[\/latex] is completely different from what happens at points close to [latex]x=7[\/latex] on either side. The notation\r\n<div style=\"text-align: center;\">[latex]\\underset{x\\to 7}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]<\/div>\r\nindicates that as the input [latex]x[\/latex] approaches 7 from either the left or the right, the output approaches 8. The output can get as close to 8 as we like if the input is sufficiently near 7.\r\n\r\nWhat happens at [latex]x=7?[\/latex] When [latex]x=7[\/latex], there is no corresponding output. We write this as\r\n<div style=\"text-align: center;\">[latex]f\\left(7\\right)\\text{ does not exist}\\text{.}[\/latex]<\/div>\r\nThis notation indicates that 7 is not in the domain of the function. We had already indicated this when we wrote the function as\r\n<div style=\"text-align: center;\">[latex]f\\left(x\\right)=x+1,\\text{ }x\\ne 7[\/latex].<\/div>\r\nNotice that the limit of a function can exist even when [latex]f\\left(x\\right)[\/latex] is not defined at [latex]x=a[\/latex]. Much of our subsequent work will be determining limits of functions as [latex]x[\/latex] nears [latex]a[\/latex], even though the output at [latex]x=a[\/latex] does not exist.\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>the limit of a function<\/h3>\r\nA quantity [latex]L[\/latex] is the <strong>limit<\/strong> of a function [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] if, as the input values of [latex]x[\/latex] approach [latex]a[\/latex] (but do not equal [latex]a[\/latex]), the corresponding output values of [latex]f\\left(x\\right)[\/latex] get closer to [latex]L[\/latex]. Note that the value of the limit is not affected by the output value of [latex]f\\left(x\\right)[\/latex] at [latex]a[\/latex]. Both [latex]a[\/latex] and [latex]L[\/latex] must be real numbers. We write it as\r\n<p style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">For the following limit, define [latex]a,f\\left(x\\right)[\/latex], and [latex]L[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(3x+5\\right)=11[\/latex]<\/p>\r\n[reveal-answer q=\"987218\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"987218\"]\r\n\r\nFirst, we recognize the notation of a limit. If the limit exists, as [latex]x[\/latex] approaches [latex]a[\/latex], we write\r\n<p style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/p>\r\nWe are given\r\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(3x+5\\right)=11[\/latex].<\/p>\r\nThis means that [latex]a=2,f\\left(x\\right)=3x+5,\\text{ and }L=11[\/latex].\r\n\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nRecall that [latex]y=3x+5[\/latex] is a line with no breaks. As the input values approach 2, the output values will get close to 11. This may be phrased with the equation [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(3x+5\\right)=11[\/latex], which means that as [latex]x[\/latex] nears 2 (but is not exactly 2), the output of the function [latex]f\\left(x\\right)=3x+5[\/latex] gets as close as we want to [latex]3\\left(2\\right)+5[\/latex], or 11, which is the limit [latex]L[\/latex], as we take values of [latex]x[\/latex] sufficiently near 2 but not at [latex]x=2[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"bcc-box bcc-success\">\r\n\r\nFor the following limit, define [latex]a,f\\left(x\\right)[\/latex], and [latex]L[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 5}{\\mathrm{lim}}\\left(2{x}^{2}-4\\right)=46[\/latex]<\/p>\r\n[reveal-answer q=\"792719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"792719\"]\r\n\r\n[latex]a=5[\/latex], [latex]f\\left(x\\right)=2{x}^{2}-4[\/latex], and [latex]L=46[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>\r\n<dl id=\"fs-id1165137641237\" class=\"definition\">\r\n \t<dd id=\"fs-id1165137641242\"><\/dd>\r\n<\/dl>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Find a limit using a graph.<\/li>\n<li style=\"font-weight: 400;\">Find a limit using a table.<\/li>\n<\/ul>\n<\/section>\n<h2>Understanding Limit Notation<\/h2>\n<p>We have seen how a <strong>sequence<\/strong> can have a limit, a value that the sequence of terms moves toward as the nu mber of terms increases. For example, the terms of the sequence<\/p>\n<div style=\"text-align: center;\">[latex]1,\\frac{1}{2},\\frac{1}{4},\\frac{1}{8}..[\/latex].<\/div>\n<p>gets closer and closer to 0. A sequence is one type of function, but functions that are not sequences can also have limits. We can describe the behavior of the function as the input values get close to a specific value. If the limit of a function [latex]f\\left(x\\right)=L[\/latex], then as the input [latex]x[\/latex] gets closer and closer to [latex]a[\/latex], the output <em>y<\/em>-coordinate gets closer and closer to [latex]L[\/latex]. We say that the output &#8220;approaches&#8221; [latex]L[\/latex].<\/p>\n<p>As the input value [latex]x[\/latex] approaches [latex]a[\/latex], the output value [latex]f\\left(x\\right)[\/latex] approaches [latex]L[\/latex].<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185219\/CNX_Precalc_Figure_12_01_0012.jpg\" alt=\"Graph representing how a function with a hole at (a, L) approaches a limit.\" width=\"487\" height=\"405\" \/><figcaption class=\"wp-caption-text\">The output (y-coordinate) approaches [latex]L[\/latex] as the input (x-coordinate) approaches [latex]a[\/latex].<\/figcaption><\/figure>\n<p>We write the equation of a limit as<\/p>\n<div style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/div>\n<p>This notation indicates that as [latex]x[\/latex] approaches [latex]a[\/latex] both from the left of [latex]x=a[\/latex] and the right of [latex]x=a[\/latex], the output value approaches [latex]L[\/latex].<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Consider the function<\/p>\n<div style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{{x}^{2}-6x - 7}{x - 7}[\/latex].<\/div>\n<p>We can factor the function as shown.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}&f\\left(x\\right)=\\frac{\\cancel{\\left(x - 7\\right)}\\left(x+1\\right)}{\\cancel{x - 7}}&& \\text{Cancel like factors in numerator and denominator.} \\\\ &f\\left(x\\right)=x+1,x\\ne 7&& \\text{Simplify}. \\end{align}[\/latex]<\/div>\n<p>Notice that [latex]x[\/latex] cannot be 7, or we would be dividing by 0, so 7 is not in the domain of the original function. In order to avoid changing the function when we simplify, we set the same condition, [latex]x\\ne 7[\/latex], for the simplified function. We can represent the function graphically.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185222\/CNX_Precalc_Figure_12_01_0022.jpg\" alt=\"Graph of an increasing function, f(x) = (x^2-6x-7)\/(x-7), with a hole at (7, 8).\" width=\"487\" height=\"483\" \/><figcaption class=\"wp-caption-text\">Because 7 is not allowed as an input, there is no point at [latex]x=7[\/latex].<\/figcaption><\/figure>\n<p>What happens at [latex]x=7[\/latex] is completely different from what happens at points close to [latex]x=7[\/latex] on either side. The notation<\/p>\n<div style=\"text-align: center;\">[latex]\\underset{x\\to 7}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]<\/div>\n<p>indicates that as the input [latex]x[\/latex] approaches 7 from either the left or the right, the output approaches 8. The output can get as close to 8 as we like if the input is sufficiently near 7.<\/p>\n<p>What happens at [latex]x=7?[\/latex] When [latex]x=7[\/latex], there is no corresponding output. We write this as<\/p>\n<div style=\"text-align: center;\">[latex]f\\left(7\\right)\\text{ does not exist}\\text{.}[\/latex]<\/div>\n<p>This notation indicates that 7 is not in the domain of the function. We had already indicated this when we wrote the function as<\/p>\n<div style=\"text-align: center;\">[latex]f\\left(x\\right)=x+1,\\text{ }x\\ne 7[\/latex].<\/div>\n<p>Notice that the limit of a function can exist even when [latex]f\\left(x\\right)[\/latex] is not defined at [latex]x=a[\/latex]. Much of our subsequent work will be determining limits of functions as [latex]x[\/latex] nears [latex]a[\/latex], even though the output at [latex]x=a[\/latex] does not exist.<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>the limit of a function<\/h3>\n<p>A quantity [latex]L[\/latex] is the <strong>limit<\/strong> of a function [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] if, as the input values of [latex]x[\/latex] approach [latex]a[\/latex] (but do not equal [latex]a[\/latex]), the corresponding output values of [latex]f\\left(x\\right)[\/latex] get closer to [latex]L[\/latex]. Note that the value of the limit is not affected by the output value of [latex]f\\left(x\\right)[\/latex] at [latex]a[\/latex]. Both [latex]a[\/latex] and [latex]L[\/latex] must be real numbers. We write it as<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">For the following limit, define [latex]a,f\\left(x\\right)[\/latex], and [latex]L[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(3x+5\\right)=11[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q987218\">Show Solution<\/button><\/p>\n<div id=\"q987218\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we recognize the notation of a limit. If the limit exists, as [latex]x[\/latex] approaches [latex]a[\/latex], we write<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/p>\n<p>We are given<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(3x+5\\right)=11[\/latex].<\/p>\n<p>This means that [latex]a=2,f\\left(x\\right)=3x+5,\\text{ and }L=11[\/latex].<\/p>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>Recall that [latex]y=3x+5[\/latex] is a line with no breaks. As the input values approach 2, the output values will get close to 11. This may be phrased with the equation [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(3x+5\\right)=11[\/latex], which means that as [latex]x[\/latex] nears 2 (but is not exactly 2), the output of the function [latex]f\\left(x\\right)=3x+5[\/latex] gets as close as we want to [latex]3\\left(2\\right)+5[\/latex], or 11, which is the limit [latex]L[\/latex], as we take values of [latex]x[\/latex] sufficiently near 2 but not at [latex]x=2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"bcc-box bcc-success\">\n<p>For the following limit, define [latex]a,f\\left(x\\right)[\/latex], and [latex]L[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 5}{\\mathrm{lim}}\\left(2{x}^{2}-4\\right)=46[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q792719\">Show Solution<\/button><\/p>\n<div id=\"q792719\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]a=5[\/latex], [latex]f\\left(x\\right)=2{x}^{2}-4[\/latex], and [latex]L=46[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<dl id=\"fs-id1165137641237\" class=\"definition\">\n<dd id=\"fs-id1165137641242\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":263,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/264"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/264\/revisions"}],"predecessor-version":[{"id":4879,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/264\/revisions\/4879"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/263"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/264\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=264"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=264"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=264"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=264"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}