{"id":1440,"date":"2024-04-17T14:02:12","date_gmt":"2024-04-17T14:02:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1440"},"modified":"2024-08-05T12:21:07","modified_gmt":"2024-08-05T12:21:07","slug":"introduction-to-the-limit-of-a-function-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/introduction-to-the-limit-of-a-function-learn-it-2\/","title":{"raw":"Introduction to the Limit of a Function: Learn It 2","rendered":"Introduction to the Limit of a Function: Learn It 2"},"content":{"raw":"<h2>The Definition of a Limit Cont.<\/h2>\r\n<h3>Estimating Limits Using Graphs<\/h3>\r\n<p id=\"fs-id1170572506486\">At this point, we see from the tables\u00a0that it may be just as easy, if not easier, to estimate a limit of a function by inspecting its graph as it is to estimate the limit by using a table of functional values.<\/p>\r\n<p>In the example below, we evaluate a limit exclusively by looking at a graph rather than by using a table of functional values.\u00a0<\/p>\r\n<section class=\"textbox recall\">\r\n<p>When looking at a graph, a function's value at a given [latex]x[\/latex] value is simply the [latex]y[\/latex] value at [latex]x[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572347401\">For [latex]g(x)[\/latex] shown in the figure below, evaluate [latex]\\underset{x\\to -1}{\\lim}g(x)[\/latex].<\/p>\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\/2332\/2018\/01\/11202858\/CNX_Calc_Figure_02_02_006.jpg\" alt=\"The graph of a generic curving function g(x). In quadrant two, there is an open circle on the function at (-1,3) and a closed circle one unit up at (-1, 4).\" width=\"487\" height=\"390\" \/> Figure 4. The graph of [latex]g(x)[\/latex] includes one value not on a smooth curve.[\/caption]\r\n[reveal-answer q=\"fs-id1170571654410\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170571654410\"]\r\n\r\n<p id=\"fs-id1170571654410\">Despite the fact that [latex]g(-1)=4[\/latex], as the [latex]x[\/latex]-values approach [latex]\u22121[\/latex] from either side, the [latex]g(x)[\/latex] values approach [latex]3[\/latex]. Therefore, [latex]\\underset{x\\to -1}{\\lim}g(x)=3[\/latex]. <br \/>\r\n<br \/>\r\n<em>Note that we can determine this limit without even knowing the algebraic expression of the function.<\/em><\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p id=\"fs-id1170571654758\">Based on the example above, we make the following observation: It is possible for the limit of a function to exist at a point, and for the function to be defined at this point, but the limit of the function and the value of the function at the point may be different.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>[caption]Watch the following video to see more examples of evaluating a limit using a graph.[\/caption]<\/p>\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qiHi41CfnFA?controls=0&amp;start=329&amp;end=405&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p>You can view the transcript for this video using <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.2TheLimitOfAFunction329to405_transcript.txt\" target=\"_blank\" rel=\"noopener\">this link<\/a> (opens in new window).[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h3>Exploring the Limits: Beyond Tables and Graphs<\/h3>\r\n<p id=\"fs-id1170572086316\">Analyzing functional values in tables or observing a function's graph can offer initial insights into its limit at a particular point. However, these approaches often involve a degree of estimation.<\/p>\r\n<p>To gain a more precise understanding, we'll move towards algebraic methods for evaluating limits. Before we delve into these methods in the following section, we'll introduce two essential limits that underpin the forthcoming algebraic techniques.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>basic limit properties<\/h3>\r\n<p id=\"fs-id1170572243382\">Let [latex]a[\/latex] be a real number and [latex]c[\/latex] be a constant.<\/p>\r\n<ol id=\"fs-id1170571659112\">\r\n\t<li>\r\n<div id=\"fs-id1170571611919\" class=\"equation\">[latex]\\underset{x\\to a}{\\lim}x=a[\/latex]<\/div>\r\n<\/li>\r\n\t<li>\r\n<div id=\"fs-id1170571600104\" class=\"equation\">[latex]\\underset{x\\to a}{\\lim}c=c[\/latex]<\/div>\r\n<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p id=\"fs-id1170571655925\">We can make the following observations about these two limits.<\/p>\r\n<ol id=\"fs-id1170572305900\">\r\n\t<li>For the first limit, observe that as [latex]x[\/latex] approaches [latex]a[\/latex], so does [latex]f(x)[\/latex], because [latex]f(x)=x[\/latex]. Consequently, [latex]\\underset{x\\to a}{\\lim}x=a[\/latex].<\/li>\r\n\t<li>For the second limit, consider the table below.<\/li>\r\n<\/ol>\r\n<table id=\"fs-id1170571613026\" summary=\"Two tables side by side, both containing two columns and five rows. The first table has headers x and f(x) = c in the first row. Under x in the first column are the values a-0.1, a-0.01, a-0.001, and a-0.0001. All of the values in the second column under the header are c. The second table has the same headers. Under x in the first column are the values a+0.1, a+0.01, a+0.001, and a+0.0001. All of the values in the second column under the header are c.\">\r\n<caption>Table of Functional Values for [latex]\\underset{x\\to a}{\\lim}c=c[\/latex]<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)=c[\/latex]<\/th>\r\n<th>\u00a0<\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)=c[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]a-0.1[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td rowspan=\"4\">\u00a0<\/td>\r\n<td>[latex]a+0.1[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a-0.01[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex]a+0.01[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a-0.001[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex]a+0.001[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a-0.0001[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex]a+0.0001[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170571576778\">Observe that for all values of [latex]x[\/latex] (regardless of whether they are approaching [latex]a[\/latex]), the values [latex]f(x)[\/latex] remain constant at [latex]c[\/latex]. We have no choice but to conclude [latex]\\underset{x\\to a}{\\lim}c=c[\/latex].<\/p>\r\n<div id=\"fs-id1170572342287\" class=\"bc-section section\">\r\n<h3>The Existence of a Limit<\/h3>\r\n<p id=\"fs-id1170572342292\">Understanding when a limit exists is fundamental to grasping the behavior of functions as their inputs approach a specific value.<\/p>\r\n<p>A limit is considered to exist at a certain point if the function values converge to a single, real number as we near that point from any direction. This behavior is indicative of the function's stability near the point of interest.<\/p>\r\n<p>If, instead, the function values diverge or oscillate without settling on a single value, we say the limit at that point does not exist.<\/p>\r\n<p>The upcoming example illustrates this concept by examining a scenario where a limit fails to exist<\/p>\r\n<section class=\"textbox proTip\">\r\n<p>Mathematicians frequently abbreviate \u201cdoes not exist\u201d as DNE.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571656086\">Evaluate [latex]\\underset{x\\to 0}{\\lim} \\sin \\left(\\dfrac{1}{x}\\right)[\/latex] using a table of values.<\/p>\r\n<p>[reveal-answer q=\"fs-id1170571614817\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170571614817\"]<\/p>\r\n<p id=\"fs-id1170571614817\">The table below lists values for the function [latex] \\sin \\left(\\dfrac{1}{x}\\right)[\/latex] for the given values of [latex]x[\/latex].<\/p>\r\n<table id=\"fs-id1170572233784\" summary=\"Two tables side by side, each with two columns and seven rows. The headers are the same, x and sin(1\/x) in the first row. In the first table, the values in the first column under x are -0.1, -0.01, -0.001, -0.0001, -0.00001, and -0.000001. The values in the second column under the header are 0.544021110889, 0.50636564111, \u22120;.8268795405312, 0.305614388888, \u22120;.035748797987, and 0.349993504187. In the second column, the values in the first column under x are 0.1, 0.01, 0.001, 0.0001, 0.00001, and 0.000001. The values in the second column under the header are \u22120;.544021110889, \u22120;.50636564111, 0.826879540532, \u22120;.305614388888, 0.035748797987, and \u22120;.349993504187.\">\r\n<caption>Table of Functional Values for [latex]\\underset{x\\to 0}{\\lim} \\sin (\\frac{1}{x})[\/latex]<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex] \\sin (\\frac{1}{x})[\/latex]<\/th>\r\n<th>\u00a0<\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex] \\sin (\\frac{1}{x})[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22120.1[\/latex]<\/td>\r\n<td>[latex]0.544021110889[\/latex]<\/td>\r\n<td rowspan=\"6\">\u00a0<\/td>\r\n<td>[latex]0.1[\/latex]<\/td>\r\n<td>[latex]\u22120.544021110889[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22120.01[\/latex]<\/td>\r\n<td>[latex]0.50636564111[\/latex]<\/td>\r\n<td>[latex]0.01[\/latex]<\/td>\r\n<td>[latex]\u22120.50636564111[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22120.001[\/latex]<\/td>\r\n<td>[latex]\u22120.8268795405312[\/latex]<\/td>\r\n<td>[latex]0.001[\/latex]<\/td>\r\n<td>[latex]0.826879540532[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22120.0001[\/latex]<\/td>\r\n<td>[latex]0.305614388888[\/latex]<\/td>\r\n<td>[latex]0.0001[\/latex]<\/td>\r\n<td>[latex]\u22120.305614388888[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22120.00001[\/latex]<\/td>\r\n<td>[latex]\u22120.035748797987[\/latex]<\/td>\r\n<td>[latex]0.00001[\/latex]<\/td>\r\n<td>[latex]0.035748797987[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\u22120.000001[\/latex]<\/td>\r\n<td>[latex]0.349993504187[\/latex]<\/td>\r\n<td>[latex]0.000001[\/latex]<\/td>\r\n<td>[latex]\u22120.349993504187[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170572420238\">After examining the table of functional values, we can see that the [latex]y[\/latex]-values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let\u2019s take a more systematic approach. Take the following sequence of [latex]x[\/latex]-values approaching 0:<\/p>\r\n<div id=\"fs-id1170572420254\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{2}{\\pi }, \\, \\frac{2}{3\\pi }, \\, \\frac{2}{5\\pi }, \\, \\frac{2}{7\\pi }, \\, \\frac{2}{9\\pi }, \\, \\frac{2}{11\\pi }, \\, \\cdots[\/latex]<\/div>\r\n<p id=\"fs-id1170572561333\">The corresponding [latex]y[\/latex]-values are<\/p>\r\n<div id=\"fs-id1170572561341\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1, \\, -1, \\, 1, \\, -1, \\, 1, \\, -1, \\, \\cdots[\/latex]<\/div>\r\n<p id=\"fs-id1170571594790\">At this point we can indeed conclude that [latex]\\underset{x\\to 0}{\\lim} \\sin (\\frac{1}{x})[\/latex] does not exist.<\/p>\r\n<p>Thus, we would write [latex]\\underset{x\\to 0}{\\lim} \\sin (\\frac{1}{x})[\/latex] DNE.<\/p>\r\n<p>The graph of [latex]f(x)= \\sin (\\frac{1}{x})[\/latex] is shown in Figure 6 and it gives a clearer picture of the behavior of [latex] \\sin (\\frac{1}{x})[\/latex] as [latex]x[\/latex] approaches [latex]0[\/latex]. You can see that [latex] \\sin (\\frac{1}{x})[\/latex] oscillates ever more wildly between [latex]\u22121[\/latex] and [latex]1[\/latex] as [latex]x[\/latex] approaches [latex]0[\/latex].<\/p>\r\n\r\n[caption id=\"attachment_1456\" align=\"alignnone\" width=\"480\"]<img class=\"wp-image-1456 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/17142405\/Screenshot-2024-04-17-102342.png\" alt=\"The graph of the function f(x) = sin(1\/x), which oscillates rapidly between -1 and 1 as x approaches 0. The oscillations are less frequent as the function moves away from 0 on the x axis.\" width=\"480\" height=\"353\" \/> Figure 6. The graph of [latex]f(x)= \\sin (\\frac{1}{x})[\/latex] oscillates rapidly between \u22121 and 1 as x approaches 0.[\/caption]\r\n[\/hidden-answer]<\/section>\r\n<\/div>\r\n<section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]204160[\/ohm_question]<\/p>\r\n<\/section>\r\n<\/section>","rendered":"<h2>The Definition of a Limit Cont.<\/h2>\n<h3>Estimating Limits Using Graphs<\/h3>\n<p id=\"fs-id1170572506486\">At this point, we see from the tables\u00a0that it may be just as easy, if not easier, to estimate a limit of a function by inspecting its graph as it is to estimate the limit by using a table of functional values.<\/p>\n<p>In the example below, we evaluate a limit exclusively by looking at a graph rather than by using a table of functional values.\u00a0<\/p>\n<section class=\"textbox recall\">\n<p>When looking at a graph, a function&#8217;s value at a given [latex]x[\/latex] value is simply the [latex]y[\/latex] value at [latex]x[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572347401\">For [latex]g(x)[\/latex] shown in the figure below, evaluate [latex]\\underset{x\\to -1}{\\lim}g(x)[\/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\/2332\/2018\/01\/11202858\/CNX_Calc_Figure_02_02_006.jpg\" alt=\"The graph of a generic curving function g(x). In quadrant two, there is an open circle on the function at (-1,3) and a closed circle one unit up at (-1, 4).\" width=\"487\" height=\"390\" \/><figcaption class=\"wp-caption-text\">Figure 4. The graph of [latex]g(x)[\/latex] includes one value not on a smooth curve.<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571654410\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571654410\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571654410\">Despite the fact that [latex]g(-1)=4[\/latex], as the [latex]x[\/latex]-values approach [latex]\u22121[\/latex] from either side, the [latex]g(x)[\/latex] values approach [latex]3[\/latex]. Therefore, [latex]\\underset{x\\to -1}{\\lim}g(x)=3[\/latex]. <\/p>\n<p><em>Note that we can determine this limit without even knowing the algebraic expression of the function.<\/em><\/p>\n<\/div>\n<\/div>\n<\/section>\n<p id=\"fs-id1170571654758\">Based on the example above, we make the following observation: It is possible for the limit of a function to exist at a point, and for the function to be defined at this point, but the limit of the function and the value of the function at the point may be different.<\/p>\n<section class=\"textbox watchIt\">\nWatch the following video to see more examples of evaluating a limit using a graph.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qiHi41CfnFA?controls=0&amp;start=329&amp;end=405&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/button><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the transcript for this video using <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.2TheLimitOfAFunction329to405_transcript.txt\" target=\"_blank\" rel=\"noopener\">this link<\/a> (opens in new window).<\/div>\n<\/div>\n<\/section>\n<h3>Exploring the Limits: Beyond Tables and Graphs<\/h3>\n<p id=\"fs-id1170572086316\">Analyzing functional values in tables or observing a function&#8217;s graph can offer initial insights into its limit at a particular point. However, these approaches often involve a degree of estimation.<\/p>\n<p>To gain a more precise understanding, we&#8217;ll move towards algebraic methods for evaluating limits. Before we delve into these methods in the following section, we&#8217;ll introduce two essential limits that underpin the forthcoming algebraic techniques.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>basic limit properties<\/h3>\n<p id=\"fs-id1170572243382\">Let [latex]a[\/latex] be a real number and [latex]c[\/latex] be a constant.<\/p>\n<ol id=\"fs-id1170571659112\">\n<li>\n<div id=\"fs-id1170571611919\" class=\"equation\">[latex]\\underset{x\\to a}{\\lim}x=a[\/latex]<\/div>\n<\/li>\n<li>\n<div id=\"fs-id1170571600104\" class=\"equation\">[latex]\\underset{x\\to a}{\\lim}c=c[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<p id=\"fs-id1170571655925\">We can make the following observations about these two limits.<\/p>\n<ol id=\"fs-id1170572305900\">\n<li>For the first limit, observe that as [latex]x[\/latex] approaches [latex]a[\/latex], so does [latex]f(x)[\/latex], because [latex]f(x)=x[\/latex]. Consequently, [latex]\\underset{x\\to a}{\\lim}x=a[\/latex].<\/li>\n<li>For the second limit, consider the table below.<\/li>\n<\/ol>\n<table id=\"fs-id1170571613026\" summary=\"Two tables side by side, both containing two columns and five rows. The first table has headers x and f(x) = c in the first row. Under x in the first column are the values a-0.1, a-0.01, a-0.001, and a-0.0001. All of the values in the second column under the header are c. The second table has the same headers. Under x in the first column are the values a+0.1, a+0.01, a+0.001, and a+0.0001. All of the values in the second column under the header are c.\">\n<caption>Table of Functional Values for [latex]\\underset{x\\to a}{\\lim}c=c[\/latex]<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)=c[\/latex]<\/th>\n<th>\u00a0<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)=c[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]a-0.1[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td rowspan=\"4\">\u00a0<\/td>\n<td>[latex]a+0.1[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a-0.01[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]a+0.01[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a-0.001[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]a+0.001[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a-0.0001[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]a+0.0001[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170571576778\">Observe that for all values of [latex]x[\/latex] (regardless of whether they are approaching [latex]a[\/latex]), the values [latex]f(x)[\/latex] remain constant at [latex]c[\/latex]. We have no choice but to conclude [latex]\\underset{x\\to a}{\\lim}c=c[\/latex].<\/p>\n<div id=\"fs-id1170572342287\" class=\"bc-section section\">\n<h3>The Existence of a Limit<\/h3>\n<p id=\"fs-id1170572342292\">Understanding when a limit exists is fundamental to grasping the behavior of functions as their inputs approach a specific value.<\/p>\n<p>A limit is considered to exist at a certain point if the function values converge to a single, real number as we near that point from any direction. This behavior is indicative of the function&#8217;s stability near the point of interest.<\/p>\n<p>If, instead, the function values diverge or oscillate without settling on a single value, we say the limit at that point does not exist.<\/p>\n<p>The upcoming example illustrates this concept by examining a scenario where a limit fails to exist<\/p>\n<section class=\"textbox proTip\">\n<p>Mathematicians frequently abbreviate \u201cdoes not exist\u201d as DNE.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571656086\">Evaluate [latex]\\underset{x\\to 0}{\\lim} \\sin \\left(\\dfrac{1}{x}\\right)[\/latex] using a table of values.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571614817\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571614817\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571614817\">The table below lists values for the function [latex]\\sin \\left(\\dfrac{1}{x}\\right)[\/latex] for the given values of [latex]x[\/latex].<\/p>\n<table id=\"fs-id1170572233784\" summary=\"Two tables side by side, each with two columns and seven rows. The headers are the same, x and sin(1\/x) in the first row. In the first table, the values in the first column under x are -0.1, -0.01, -0.001, -0.0001, -0.00001, and -0.000001. The values in the second column under the header are 0.544021110889, 0.50636564111, \u22120;.8268795405312, 0.305614388888, \u22120;.035748797987, and 0.349993504187. In the second column, the values in the first column under x are 0.1, 0.01, 0.001, 0.0001, 0.00001, and 0.000001. The values in the second column under the header are \u22120;.544021110889, \u22120;.50636564111, 0.826879540532, \u22120;.305614388888, 0.035748797987, and \u22120;.349993504187.\">\n<caption>Table of Functional Values for [latex]\\underset{x\\to 0}{\\lim} \\sin (\\frac{1}{x})[\/latex]<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\sin (\\frac{1}{x})[\/latex]<\/th>\n<th>\u00a0<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\sin (\\frac{1}{x})[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\u22120.1[\/latex]<\/td>\n<td>[latex]0.544021110889[\/latex]<\/td>\n<td rowspan=\"6\">\u00a0<\/td>\n<td>[latex]0.1[\/latex]<\/td>\n<td>[latex]\u22120.544021110889[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u22120.01[\/latex]<\/td>\n<td>[latex]0.50636564111[\/latex]<\/td>\n<td>[latex]0.01[\/latex]<\/td>\n<td>[latex]\u22120.50636564111[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u22120.001[\/latex]<\/td>\n<td>[latex]\u22120.8268795405312[\/latex]<\/td>\n<td>[latex]0.001[\/latex]<\/td>\n<td>[latex]0.826879540532[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u22120.0001[\/latex]<\/td>\n<td>[latex]0.305614388888[\/latex]<\/td>\n<td>[latex]0.0001[\/latex]<\/td>\n<td>[latex]\u22120.305614388888[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u22120.00001[\/latex]<\/td>\n<td>[latex]\u22120.035748797987[\/latex]<\/td>\n<td>[latex]0.00001[\/latex]<\/td>\n<td>[latex]0.035748797987[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\u22120.000001[\/latex]<\/td>\n<td>[latex]0.349993504187[\/latex]<\/td>\n<td>[latex]0.000001[\/latex]<\/td>\n<td>[latex]\u22120.349993504187[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170572420238\">After examining the table of functional values, we can see that the [latex]y[\/latex]-values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let\u2019s take a more systematic approach. Take the following sequence of [latex]x[\/latex]-values approaching 0:<\/p>\n<div id=\"fs-id1170572420254\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{2}{\\pi }, \\, \\frac{2}{3\\pi }, \\, \\frac{2}{5\\pi }, \\, \\frac{2}{7\\pi }, \\, \\frac{2}{9\\pi }, \\, \\frac{2}{11\\pi }, \\, \\cdots[\/latex]<\/div>\n<p id=\"fs-id1170572561333\">The corresponding [latex]y[\/latex]-values are<\/p>\n<div id=\"fs-id1170572561341\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1, \\, -1, \\, 1, \\, -1, \\, 1, \\, -1, \\, \\cdots[\/latex]<\/div>\n<p id=\"fs-id1170571594790\">At this point we can indeed conclude that [latex]\\underset{x\\to 0}{\\lim} \\sin (\\frac{1}{x})[\/latex] does not exist.<\/p>\n<p>Thus, we would write [latex]\\underset{x\\to 0}{\\lim} \\sin (\\frac{1}{x})[\/latex] DNE.<\/p>\n<p>The graph of [latex]f(x)= \\sin (\\frac{1}{x})[\/latex] is shown in Figure 6 and it gives a clearer picture of the behavior of [latex]\\sin (\\frac{1}{x})[\/latex] as [latex]x[\/latex] approaches [latex]0[\/latex]. You can see that [latex]\\sin (\\frac{1}{x})[\/latex] oscillates ever more wildly between [latex]\u22121[\/latex] and [latex]1[\/latex] as [latex]x[\/latex] approaches [latex]0[\/latex].<\/p>\n<figure id=\"attachment_1456\" aria-describedby=\"caption-attachment-1456\" style=\"width: 480px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1456 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/17142405\/Screenshot-2024-04-17-102342.png\" alt=\"The graph of the function f(x) = sin(1\/x), which oscillates rapidly between -1 and 1 as x approaches 0. The oscillations are less frequent as the function moves away from 0 on the x axis.\" width=\"480\" height=\"353\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/17142405\/Screenshot-2024-04-17-102342.png 480w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/17142405\/Screenshot-2024-04-17-102342-300x221.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/17142405\/Screenshot-2024-04-17-102342-65x48.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/17142405\/Screenshot-2024-04-17-102342-225x165.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/17142405\/Screenshot-2024-04-17-102342-350x257.png 350w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/><figcaption id=\"caption-attachment-1456\" class=\"wp-caption-text\">Figure 6. The graph of [latex]f(x)= \\sin (\\frac{1}{x})[\/latex] oscillates rapidly between \u22121 and 1 as x approaches 0.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm204160\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=204160&theme=lumen&iframe_resize_id=ohm204160&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<\/section>\n","protected":false},"author":15,"menu_order":12,"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":"learn_it","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\/1440"}],"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":20,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1440\/revisions"}],"predecessor-version":[{"id":4463,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1440\/revisions\/4463"}],"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\/1440\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1440"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1440"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1440"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1440"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}