{"id":1446,"date":"2023-06-05T14:51:49","date_gmt":"2023-06-05T14:51:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/finding-limits-numerical-and-graphical-approaches\/"},"modified":"2023-06-05T14:51:49","modified_gmt":"2023-06-05T14:51:49","slug":"finding-limits-numerical-and-graphical-approaches","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/finding-limits-numerical-and-graphical-approaches\/","title":{"raw":"Finding Limits: Numerical and Graphical Approaches","rendered":"Finding Limits: Numerical and Graphical Approaches"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li style=\"font-weight: 400;\">Find a limit using a graph.<\/li>\n \t<li style=\"font-weight: 400;\">Find a limit using a table.<\/li>\n<\/ul>\n<\/div>\n<h2>Understanding Limit Notation<\/h2>\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\n<div style=\"text-align: center;\">[latex]1,\\frac{1}{2},\\frac{1}{4},\\frac{1}{8}..[\/latex].<\/div>\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].\n\nFigure 1 provides a visual representation of the mathematical concept of limit. As the input value [latex]x[\/latex] approaches [latex]a[\/latex], the output value [latex]f\\left(x\\right)[\/latex] approaches [latex]L[\/latex].\n\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\"> <b>Figure 1.<\/b> 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\n<div style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/div>\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].\n\nConsider the function\n<div style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{{x}^{2}-6x - 7}{x - 7}[\/latex].<\/div>\nWe can factor the function as shown.\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>\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 as shown in Figure 2.\n\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\"> <b>Figure 2.<\/b> 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\n<div style=\"text-align: center;\">[latex]\\underset{x\\to 7}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]<\/div>\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.\n\nWhat happens at [latex]x=7?[\/latex] When [latex]x=7[\/latex], there is no corresponding output. We write this as\n<div style=\"text-align: center;\">[latex]f\\left(7\\right)\\text{ does not exist}\\text{.}[\/latex]<\/div>\nThis notation indicates that 7 is not in the domain of the function. We had already indicated this when we wrote the function as\n<div style=\"text-align: center;\">[latex]f\\left(x\\right)=x+1,\\text{ }x\\ne 7[\/latex].<\/div>\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.\n<div class=\"textbox\">\n<h3>A General Note: The Limit of a Function<\/h3>\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\n<p style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Understanding the Limit of a Function<\/h3>\nFor the following limit, define [latex]a,f\\left(x\\right)[\/latex], and [latex]L[\/latex].\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(3x+5\\right)=11[\/latex]<\/p>\n[reveal-answer q=\"987218\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"987218\"]\n\nFirst, we recognize the notation of a limit. If the limit exists, as [latex]x[\/latex] approaches [latex]a[\/latex], we write\n<p style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/p>\nWe are given\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(3x+5\\right)=11[\/latex].<\/p>\nThis means that [latex]a=2,f\\left(x\\right)=3x+5,\\text{ and }L=11[\/latex].\n<h4>Analysis of the Solution<\/h4>\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].\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nFor the following limit, define [latex]a,f\\left(x\\right)[\/latex], and [latex]L[\/latex].\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 5}{\\mathrm{lim}}\\left(2{x}^{2}-4\\right)=46[\/latex]<\/p>\n[reveal-answer q=\"792719\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"792719\"]\n\n[latex]a=5[\/latex], [latex]f\\left(x\\right)=2{x}^{2}-4[\/latex], and [latex]L=46[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\n<h2>Understanding Left-Hand Limits and Right-Hand Limits<\/h2>\nWe can approach the input of a function from either side of a value\u2014from the left or the right. Figure 3&nbsp;shows the values of\n<div style=\"text-align: center;\">[latex]f\\left(x\\right)=x+1,x\\ne 7[\/latex]<\/div>\nas described earlier and depicted in Figure 2.\n\n[caption id=\"\" align=\"alignnone\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185224\/CNX_Precalc_Figure_12_01_0132.jpg\" alt=\"Table showing that f(x) approaches 8 from either side as x approaches 7 from either side.\" width=\"975\" height=\"180\"> <b>Figure 3<\/b>[\/caption]\n\nValues described as \"from the left\" are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left in Figure 3&nbsp;are [latex]6.9[\/latex], [latex]6.99[\/latex], and [latex]6.999[\/latex]. The corresponding outputs are [latex]7.9,7.99[\/latex], and [latex]7.999[\/latex]. These values are getting closer to 8. The limit of values of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function [latex]f\\left(x\\right)=x+1,x\\ne 7[\/latex] as [latex]x[\/latex] approaches 7.\n\nValues described as \"from the right\" are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right in&nbsp;Figure 3&nbsp;are [latex]7.1[\/latex], [latex]7.01[\/latex], and [latex]7.001[\/latex]. The corresponding outputs are [latex]8.1[\/latex], [latex]8.01[\/latex], and [latex]8.001[\/latex]. These values are getting closer to 8. The limit of values of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function [latex]f\\left(x\\right)=x+1,x\\ne 7[\/latex] as [latex]x[\/latex] approaches 7.\n\nFigure 3 shows that we can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input [latex]x[\/latex] within the interval [latex]6.9&lt;x&lt;7.1[\/latex] to produce an output value of [latex]f\\left(x\\right)[\/latex] within the interval [latex]7.9&lt;f\\left(x\\right)&lt;8.1[\/latex].\n\nWe also see that we can get output values of [latex]f\\left(x\\right)[\/latex] successively closer to 8 by selecting input values closer to 7. In fact, we can obtain output values within any specified interval if we choose appropriate input values.\n\nFigure 4 provides a visual representation of the left- and right-hand limits of the function. From the graph of [latex]f\\left(x\\right)[\/latex], we observe the output can get infinitesimally close to [latex]L=8[\/latex] as [latex]x[\/latex] approaches 7 from the left and as [latex]x[\/latex] approaches 7 from the right.\n\nTo indicate the left-hand limit, we write\n<div style=\"text-align: center;\">[latex]\\underset{x\\to {7}^{-}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex].<\/div>\nTo indicate the right-hand limit, we write\n<div style=\"text-align: center;\">[latex]\\underset{x\\to {7}^{+}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex].<\/div>\n\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\/27185226\/CNX_Precalc_Figure_12_01_0032.jpg\" alt=\"Graph of the previous function explaining the function's limit at (7, 8)\" width=\"487\" height=\"483\"> <b>Figure 4.<\/b> The left- and right-hand limits are the same for this function.[\/caption]\n\n<div class=\"textbox\">\n<h3>A General Note: Left- and Right-Hand Limits<\/h3>\nThe <strong>left-hand limit<\/strong> of a function [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the left is equal to [latex]L[\/latex], denoted by\n<p style=\"text-align: center;\">[latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/p>\nThe values of [latex]f\\left(x\\right)[\/latex] can get as close to the limit [latex]L[\/latex] as we like by taking values of [latex]x[\/latex] sufficiently close to [latex]a[\/latex] such that [latex]x&lt;a[\/latex] and [latex]x\\ne a[\/latex].\n\nThe <strong>right-hand limit<\/strong> of a function [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]a[\/latex] from the right, is equal to [latex]L[\/latex], denoted by\n<p style=\"text-align: center;\">[latex]\\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/p>\nThe values of [latex]f\\left(x\\right)[\/latex] can get as close to the limit [latex]L[\/latex] as we like by taking values of [latex]x[\/latex] sufficiently close to [latex]a[\/latex] but greater than [latex]a[\/latex]. Both [latex]a[\/latex] and [latex]L[\/latex] are real numbers.\n\n<\/div>\n<h2>Understanding Two-Sided Limits<\/h2>\nIn the previous example, the left-hand limit and right-hand limit as [latex]x[\/latex] approaches [latex]a[\/latex] are equal. If the left- and right-hand limits are equal, we say that the function [latex]f\\left(x\\right)[\/latex] has a <strong>two-sided limit<\/strong> as [latex]x[\/latex] approaches [latex]a[\/latex]. More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.\n<div class=\"textbox\">\n<h3>A General Note: The Two-Sided Limit of Function as <em>x<\/em> Approaches <em>a<\/em><\/h3>\nThe limit of a function [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]a[\/latex], is equal to [latex]L[\/latex], that is,\n<p style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex]<\/p>\nif and only if\n<p style=\"text-align: center;\">[latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex].<\/p>\nIn other words, the left-hand limit of a function [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is equal to the right-hand limit of the same function as [latex]x[\/latex] approaches [latex]a[\/latex]. If such a limit exists, we refer to the limit as a two-sided limit. Otherwise we say the limit does not exist.\n\n<\/div>\n<h2>Finding a Limit Using a Graph<\/h2>\nTo visually determine if a limit exists as [latex]x[\/latex] approaches [latex]a[\/latex], we observe the graph of the function when [latex]x[\/latex] is very near to [latex]x=a[\/latex]. In Figure 5&nbsp;we observe the behavior of the graph on both sides of [latex]a[\/latex].\n\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\/27185228\/CNX_Precalc_Figure_12_01_004F2.jpg\" alt=\"Graph of a function that explains the behavior of a limit at (a, L) where the function is increasing when x is less than a and decreasing when x is greater than a.\" width=\"487\" height=\"408\"> <b>Figure 5<\/b>[\/caption]\n\nTo determine if a left-hand limit exists, we observe the branch of the graph to the left of [latex]x=a[\/latex], but near [latex]x=a[\/latex]. This is where [latex]x&lt;a[\/latex]. We see that the outputs are getting close to some real number [latex]L[\/latex] so there is a left-hand limit.\n\nTo determine if a right-hand limit exists, observe the branch of the graph to the right of [latex]x=a[\/latex], but near [latex]x=a[\/latex]. This is where [latex]x&gt;a[\/latex]. We see that the outputs are getting close to some real number [latex]L[\/latex], so there is a right-hand limit.\n\nIf the left-hand limit and the right-hand limit are the same, as they are in Figure 5, then we know that the function has a two-sided limit. Normally, when we refer to a \"limit,\" we mean a two-sided limit, unless we call it a one-sided limit.\n\nFinally, we can look for an output value for the function [latex]f\\left(x\\right)[\/latex] when the input value [latex]x[\/latex] is equal to [latex]a[\/latex]. The coordinate pair of the point would be [latex]\\left(a,f\\left(a\\right)\\right)[\/latex]. If such a point exists, then [latex]f\\left(a\\right)[\/latex] has a value. If the point does not exist, as in Figure 5, then we say that [latex]f\\left(a\\right)[\/latex] does not exist.\n<div class=\"textbox\">\n<h3>How To: Given a function [latex]f\\left(x\\right)[\/latex], use a graph to find the limits and a function value as [latex]x[\/latex] approaches [latex]a[\/latex].<\/h3>\n<ol>\n \t<li>Examine the graph to determine whether a left-hand limit exists.<\/li>\n \t<li>Examine the graph to determine whether a right-hand limit exists.<\/li>\n \t<li>If the two one-sided limits exist and are equal, then there is a two-sided limit\u2014what we normally call a \"limit.\"<\/li>\n \t<li>If there is a point at [latex]x=a[\/latex], then [latex]f\\left(a\\right)[\/latex] is the corresponding function value.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Finding a Limit Using a Graph<\/h3>\n<ol>\n \t<li>Determine the following limits and function value for the function [latex]f[\/latex] shown in Figure 6.\n<ol>\n \t<li>[latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n \t<li>[latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n \t<li>[latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n \t<li>[latex]f\\left(2\\right)[\/latex]<\/li>\n<\/ol>\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\/27185230\/CNX_Precalc_Figure_12_01_0052.jpg\" alt=\"Graph of a piecewise function that has a positive parabola centered at the origin and goes from negative infinity to (2, 8), an open point, and a decreasing line from (2, 3), a closed point, to positive infinity on the x-axis.\" width=\"487\" height=\"520\"> <b>Figure 6<\/b>[\/caption]<\/li>\n \t<li>Determine the following limits and function value for the function [latex]f[\/latex] shown in Figure 7.\n<ol>\n \t<li>[latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n \t<li>[latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n \t<li>[latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n \t<li>[latex]f\\left(2\\right)[\/latex]<\/li>\n<\/ol>\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\/27185232\/CNX_Precalc_Figure_12_01_0062.jpg\" alt=\"Graph of a piecewise function that has a positive parabola from negative infinity to 2 on the x-axis, a decreasing line from 2 to positive infinity on the x-axis, and a point at (2, 4).\" width=\"487\" height=\"483\"> <b>Figure 7<\/b>[\/caption]\n\n[reveal-answer q=\"459090\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"459090\"]\n<ol>\n \t<li>Looking at Figure 6:\n<ol>\n \t<li>[latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]; when [latex]x&lt;2[\/latex], but infinitesimally close to 2, the output values get close to [latex]y=8[\/latex].<\/li>\n \t<li>[latex]\\underset{x\\to 2{}^{+}}{\\mathrm{lim}}f\\left(x\\right)=3[\/latex]; when [latex]x&gt;2[\/latex], but infinitesimally close to 2, the output values approach [latex]y=3[\/latex].<\/li>\n \t<li>[latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex] does not exist because [latex]\\underset{x\\to 2{}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to 2{}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]; the left and right-hand limits are not equal.<\/li>\n \t<li>[latex]f\\left(2\\right)=3[\/latex] because the graph of the function [latex]f[\/latex] passes through the point [latex]\\left(2,f\\left(2\\right)\\right)[\/latex] or [latex]\\left(2,3\\right)[\/latex].<\/li>\n<\/ol>\n<\/li>\n \t<li>Looking at Figure 7:\n<ol>\n \t<li>[latex]\\underset{x\\to 2{}^{-}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]; when [latex]x&lt;2[\/latex] but infinitesimally close to 2, the output values approach [latex]y=8[\/latex].<\/li>\n \t<li>[latex]\\underset{x\\to 2{}^{+}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]; when [latex]x&gt;2[\/latex] but infinitesimally close to 2, the output values approach [latex]y=8[\/latex].<\/li>\n \t<li>[latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex] because [latex]\\underset{x\\to 2{}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to 2{}^{+}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]; the left and right-hand limits are equal.<\/li>\n \t<li>[latex]f\\left(2\\right)=4[\/latex] because the graph of the function [latex]f[\/latex] passes through the point [latex]\\left(2,f\\left(2\\right)\\right)[\/latex] or [latex]\\left(2,4\\right)[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n[\/hidden-answer]<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nUsing the graph of the function [latex]y=f\\left(x\\right)[\/latex] shown in Figure 8, estimate the following limits.\n\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\/27185234\/CNX_Precalc_Figure_12_01_007n2.jpg\" alt=\"Graph of a piecewise function that has three segments: 1) negative infinity to 0, 2) 0 to 2, and 3) 2 to positive inifnity, which has a discontinuity at (4, 4)\" width=\"487\" height=\"556\"> <b>Figure 8<\/b>[\/caption]\n\n[reveal-answer q=\"319433\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"319433\"]\n\na. 0; b. 2; c. does not exist; d. [latex]-2[\/latex]; e. 0; f. does not exist; g. 4; h. 4; i. 4\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]174077[\/ohm_question]\n\n<\/div>\n<h2>Finding a Limit Using a Table<\/h2>\nCreating a table is a way to determine limits using numeric information. We create a table of values in which the input values of [latex]x[\/latex] approach [latex]a[\/latex] from both sides. Then we determine if the output values get closer and closer to some real value, the <strong>limit<\/strong> [latex]L[\/latex].\n\nLet\u2019s consider an example using the following function:\n<div style=\"text-align: center;\">[latex]\\underset{x\\to 5}{\\mathrm{lim}}\\left(\\dfrac{{x}^{3}-125}{x - 5}\\right)[\/latex]<\/div>\nTo create the table, we evaluate the function at values close to [latex]x=5[\/latex]. We use some input values less than 5 and some values greater than 5 as in Figure 9. The table values show that when [latex]x&gt;5[\/latex] but nearing 5, the corresponding output gets close to 75. When [latex]x&gt;5[\/latex] but nearing 5, the corresponding output also gets close to 75.\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185237\/CNX_Precalc_Figure_12_01_008n2.jpg\" alt=\"Table shows that as x values approach 5 from the positive or negative direction, f(x) gets very close to 75. But when x is equal to 5, y is undefined.\" width=\"975\" height=\"137\"> <b>Figure 9<\/b>[\/caption]\n\nBecause\n<div style=\"text-align: center;\">[latex]\\underset{x\\to {5}^{-}}{\\mathrm{lim}}f\\left(x\\right)=75=\\underset{x\\to {5}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex],<\/div>\nthen\n<div style=\"text-align: center;\">[latex]\\underset{x\\to 5}{\\mathrm{lim}}f\\left(x\\right)=75[\/latex].<\/div>\nRemember that [latex]f\\left(5\\right)[\/latex] does not exist.\n<div class=\"textbox\">\n<h3>How To: Given a function [latex]f[\/latex], use a table to find the limit as [latex]x[\/latex] approaches [latex]a[\/latex] and the value of [latex]f\\left(a\\right)[\/latex], if it exists.<\/h3>\n<ol>\n \t<li>Choose several input values that approach [latex]a[\/latex] from both the left and right. Record them in a table.<\/li>\n \t<li>Evaluate the function at each input value. Record them in the table.<\/li>\n \t<li>Determine if the table values indicate a left-hand limit and a right-hand limit.<\/li>\n \t<li>If the left-hand and right-hand limits exist and are equal, there is a two-sided limit.<\/li>\n \t<li>Replace [latex]x[\/latex] with [latex]a[\/latex] to find the value of [latex]f\\left(a\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Finding a Limit Using a Table<\/h3>\nNumerically estimate the limit of the following expression by setting up a table of values on both sides of the limit.\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{5\\sin \\left(x\\right)}{3x}\\right)[\/latex]<\/p>\n[reveal-answer q=\"431562\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"431562\"]\n\nWe can estimate the value of a limit, if it exists, by evaluating the function at values near [latex]x=0[\/latex]. We cannot find a function value for [latex]x=0[\/latex] directly because the result would have a denominator equal to 0, and thus would be undefined.\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{5\\sin \\left(x\\right)}{3x}[\/latex]<\/p>\nWe create Figure 10&nbsp;by choosing several input values close to [latex]x=0[\/latex], with half of them less than [latex]x=0[\/latex] and half of them greater than [latex]x=0[\/latex]. Note that we need to be sure we are using radian mode. We evaluate the function at each input value to complete the table.\n\nThe table values indicate that when [latex]x&lt;0[\/latex] but approaching 0, the corresponding output nears [latex]\\frac{5}{3}[\/latex].\n\nWhen [latex]x&gt;0[\/latex] but approaching 0, the corresponding output also nears [latex]\\frac{5}{3}[\/latex].\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185239\/CNX_Precalc_Figure_12_01_0092.jpg\" alt=\"Table shows that as x values approach 0 from the positive or negative direction, f(x) gets very close to 5 over 3. But when x is equal to 0, y is undefined.\" width=\"975\" height=\"146\"> <b>Figure 10<\/b>[\/caption]\n\nBecause\n<p style=\"text-align: center;\">[latex]\\underset{x\\to {0}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\frac{5}{3}=\\underset{x\\to {0}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex],<\/p>\nthen\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\mathrm{lim}}f\\left(x\\right)=\\frac{5}{3}[\/latex].<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>Is it possible to check our answer using a graphing utility?<\/h3>\n<em>Yes. We previously used a table to find a limit of 75 for the function [latex]f\\left(x\\right)=\\frac{{x}^{3}-125}{x - 5}[\/latex] as [latex]x[\/latex] approaches 5. To check, we graph the function on a viewing window as shown in Figure 11. A graphical check shows both branches of the graph of the function get close to the output 75 as [latex]x[\/latex] nears 5. Furthermore, we can use the \u2018trace\u2019 feature of a graphing calculator. By appraoching [latex]x=5[\/latex] we may numerically observe the corresponding outputs getting close to [latex]75[\/latex]. <\/em>\n\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\/27185241\/CNX_Precalc_Figure_12_01_010F2.jpg\" alt=\"Graph of an increasing function with a discontinuity at (5, 75)\" width=\"487\" height=\"254\"> <b>Figure 11<\/b>[\/caption]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nNumerically estimate the limit of the following function by making a table:\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{20\\sin \\left(x\\right)}{4x}\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">[reveal-answer q=\"172208\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"172208\"]<\/p>\n[latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{20\\sin \\left(x\\right)}{4x}\\right)=5[\/latex]\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185250\/CNX_Precalc_Figure_12_01_0142.jpg\" alt=\"Table showing that f(x) approaches 5 from either side as x approaches 0 from either side.\">\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]174079[\/ohm_question]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>Is one method for determining a limit better than the other?<\/h3>\n<em>No. Both methods have advantages. Graphing allows for quick inspection. Tables can be used when graphical utilities aren\u2019t available, and they can be calculated to a higher precision than could be seen with an unaided eye inspecting a graph.<\/em>\n\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Using a Graphing Utility to Determine a Limit<\/h3>\nWith the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as [latex]x[\/latex] approaches 0. If the function has a limit as [latex]x[\/latex] approaches 0, state it. If not, discuss why there is no limit.\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=3\\sin \\left(\\frac{\\pi }{x}\\right)[\/latex]<\/p>\n[reveal-answer q=\"171628\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"171628\"]\n\nWe can use a graphing utility to investigate the behavior of the graph close to [latex]x=0[\/latex]. Centering around [latex]x=0[\/latex], we choose two viewing windows such that the second one is zoomed in closer to [latex]x=0[\/latex] than the first one. The result would resemble Figure 12&nbsp;for [latex]\\left[-2,2\\right][\/latex] by [latex]\\left[-3,3\\right][\/latex].\n\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\/27185243\/CNX_Precalc_Figure_12_01_011F2.jpg\" alt=\"Graph of a sinusodial function zoomed in at [-2, 2] by [-3, 3].\" width=\"487\" height=\"367\"> <b>Figure 12<\/b>[\/caption]The result would resemble Figure 13&nbsp;for [latex]\\left[-0.1,0.1\\right][\/latex] by [latex]\\left[-3,3\\right][\/latex].[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185245\/CNX_Precalc_Figure_12_01_012F2.jpg\" alt=\"Graph of the same sinusodial function as in the previous image zoomed in at [-0.1, 0.1] by [-3. 3].\" width=\"488\" height=\"367\"> <b>Figure 13<\/b>[\/caption]Even closer to zero, we are even less able to distinguish any limits.The closer we get to 0, the greater the swings in the output values are. That is not the behavior of a function with either a left-hand limit or a right-hand limit. And if there is no left-hand limit or right-hand limit, there certainly is no limit to the function [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches 0.We write\n<p style=\"text-align: center;\">[latex]\\underset{x\\to {0}^{-}}{\\mathrm{lim}}\\left(3\\sin \\left(\\frac{\\pi }{x}\\right)\\right)\\text{ does not exist}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to {0}^{+}}{\\mathrm{lim}}\\left(3\\sin \\left(\\frac{\\pi }{x}\\right)\\right)\\text{ does not exist}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(3\\sin \\left(\\frac{\\pi }{x}\\right)\\right)\\text{ does not exist}[\/latex].<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nNumerically estimate the following limit: [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\sin \\left(\\dfrac{2}{x}\\right)\\right)[\/latex].\n\n[reveal-answer q=\"165056\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"165056\"]\n\ndoes not exist\n\n[\/hidden-answer]\n\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>A function has a limit if the output values approach some value [latex]L[\/latex] as the input values approach some quantity [latex]a[\/latex].<\/li>\n \t<li>A shorthand notation is used to describe the limit of a function according to the form [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex], which 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 gets close to [latex]L[\/latex].<\/li>\n \t<li>A function has a left-hand limit if [latex]f\\left(x\\right)[\/latex] approaches [latex]L[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] where [latex]x&lt;a[\/latex]. A function has a right-hand limit if [latex]f\\left(x\\right)[\/latex] approaches [latex]L[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] where [latex]x&gt;a[\/latex].<\/li>\n \t<li>A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit.<\/li>\n \t<li>A graph provides a visual method of determining the limit of a function.<\/li>\n \t<li>If the function has a limit as [latex]x[\/latex] approaches [latex]a[\/latex], the branches of the graph will approach the same [latex]y\\text{-}[\/latex] coordinate near [latex]x=a[\/latex] from the left and the right.<\/li>\n \t<li>A table can be used to determine if a function has a limit. The table should show input values that approach [latex]a[\/latex] from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit.<\/li>\n \t<li>A graphing utility can also be used to find a limit.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137935677\" class=\"definition\">\n \t<dt>left-hand limit<\/dt>\n \t<dd id=\"fs-id1165137935682\">the limit of values of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches from [latex]a[\/latex] the left, denoted [latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex]. The values of [latex]f\\left(x\\right)[\/latex] can get as close to the limit [latex]L[\/latex] as we like by taking values of [latex]x[\/latex] sufficiently close to [latex]a[\/latex] such that [latex]x&lt;a[\/latex] and [latex]x\\ne a[\/latex]. Both [latex]a[\/latex] and [latex]L[\/latex] are real numbers.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137673875\" class=\"definition\">\n \t<dt>limit<\/dt>\n \t<dd id=\"fs-id1165137673880\">when it exists, the value, [latex]L[\/latex], that the output of a function [latex]f\\left(x\\right)[\/latex] approaches as the input [latex]x[\/latex] gets closer and closer to [latex]a[\/latex] but does not equal [latex]a[\/latex]. The value of the output, [latex]f\\left(x\\right)[\/latex], can get as close to [latex]L[\/latex] as we choose to make it by using input values of [latex]x[\/latex] sufficiently near to [latex]x=a[\/latex], but not necessarily at [latex]x=a[\/latex]. Both [latex]a[\/latex] and [latex]L[\/latex] are real numbers, and [latex]L[\/latex] is denoted [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832351\" class=\"definition\">\n \t<dt>right-hand limit<\/dt>\n \t<dd id=\"fs-id1165137832357\">the limit of values of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the right, denoted [latex]\\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex]. The values of [latex]f\\left(x\\right)[\/latex] can get as close to the limit [latex]L[\/latex] as we like by taking values of [latex]x[\/latex] sufficiently close to [latex]a[\/latex] where [latex]x&gt;a[\/latex], and [latex]x\\ne a[\/latex]. Both [latex]a[\/latex] and [latex]L[\/latex] are real numbers.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137641237\" class=\"definition\">\n \t<dt>two-sided limit<\/dt>\n \t<dd id=\"fs-id1165137641242\">the limit of a function [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]a[\/latex], is equal to [latex]L[\/latex], that is, [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex] if and only if [latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex].<\/dd>\n<\/dl>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\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<\/div>\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>Figure 1 provides a visual representation of the mathematical concept of limit. 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<div style=\"width: 497px\" 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\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> The output (y-coordinate) approaches [latex]L[\/latex] as the input (x-coordinate) approaches [latex]a[\/latex].<\/p>\n<\/div>\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<p>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 as shown in Figure 2.<\/p>\n<div style=\"width: 497px\" 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\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2.<\/b> Because 7 is not allowed as an input, there is no point at [latex]x=7[\/latex].<\/p>\n<\/div>\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<div class=\"textbox\">\n<h3>A General Note: 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<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Understanding the Limit of a Function<\/h3>\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 2}{\\mathrm{lim}}\\left(3x+5\\right)=11[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q987218\">Show Solution<\/span><\/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<h4>Analysis of the Solution<\/h4>\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<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q792719\">Show Solution<\/span><\/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<h2>Understanding Left-Hand Limits and Right-Hand Limits<\/h2>\n<p>We can approach the input of a function from either side of a value\u2014from the left or the right. Figure 3&nbsp;shows the values of<\/p>\n<div style=\"text-align: center;\">[latex]f\\left(x\\right)=x+1,x\\ne 7[\/latex]<\/div>\n<p>as described earlier and depicted in Figure 2.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185224\/CNX_Precalc_Figure_12_01_0132.jpg\" alt=\"Table showing that f(x) approaches 8 from either side as x approaches 7 from either side.\" width=\"975\" height=\"180\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<p>Values described as &#8220;from the left&#8221; are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left in Figure 3&nbsp;are [latex]6.9[\/latex], [latex]6.99[\/latex], and [latex]6.999[\/latex]. The corresponding outputs are [latex]7.9,7.99[\/latex], and [latex]7.999[\/latex]. These values are getting closer to 8. The limit of values of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function [latex]f\\left(x\\right)=x+1,x\\ne 7[\/latex] as [latex]x[\/latex] approaches 7.<\/p>\n<p>Values described as &#8220;from the right&#8221; are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right in&nbsp;Figure 3&nbsp;are [latex]7.1[\/latex], [latex]7.01[\/latex], and [latex]7.001[\/latex]. The corresponding outputs are [latex]8.1[\/latex], [latex]8.01[\/latex], and [latex]8.001[\/latex]. These values are getting closer to 8. The limit of values of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function [latex]f\\left(x\\right)=x+1,x\\ne 7[\/latex] as [latex]x[\/latex] approaches 7.<\/p>\n<p>Figure 3 shows that we can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input [latex]x[\/latex] within the interval [latex]6.9<x<7.1[\/latex] to produce an output value of [latex]f\\left(x\\right)[\/latex] within the interval [latex]7.9<f\\left(x\\right)<8.1[\/latex].\n\nWe also see that we can get output values of [latex]f\\left(x\\right)[\/latex] successively closer to 8 by selecting input values closer to 7. In fact, we can obtain output values within any specified interval if we choose appropriate input values.\n\nFigure 4 provides a visual representation of the left- and right-hand limits of the function. From the graph of [latex]f\\left(x\\right)[\/latex], we observe the output can get infinitesimally close to [latex]L=8[\/latex] as [latex]x[\/latex] approaches 7 from the left and as [latex]x[\/latex] approaches 7 from the right.\n\nTo indicate the left-hand limit, we write\n\n\n<div style=\"text-align: center;\">[latex]\\underset{x\\to {7}^{-}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex].<\/div>\n<p>To indicate the right-hand limit, we write<\/p>\n<div style=\"text-align: center;\">[latex]\\underset{x\\to {7}^{+}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex].<\/div>\n<div style=\"width: 497px\" 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\/27185226\/CNX_Precalc_Figure_12_01_0032.jpg\" alt=\"Graph of the previous function explaining the function's limit at (7, 8)\" width=\"487\" height=\"483\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4.<\/b> The left- and right-hand limits are the same for this function.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Left- and Right-Hand Limits<\/h3>\n<p>The <strong>left-hand limit<\/strong> of a function [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the left is equal to [latex]L[\/latex], denoted by<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/p>\n<p>The values of [latex]f\\left(x\\right)[\/latex] can get as close to the limit [latex]L[\/latex] as we like by taking values of [latex]x[\/latex] sufficiently close to [latex]a[\/latex] such that [latex]x<a[\/latex] and [latex]x\\ne a[\/latex].\n\nThe <strong>right-hand limit<\/strong> of a function [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]a[\/latex] from the right, is equal to [latex]L[\/latex], denoted by<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/p>\n<p>The values of [latex]f\\left(x\\right)[\/latex] can get as close to the limit [latex]L[\/latex] as we like by taking values of [latex]x[\/latex] sufficiently close to [latex]a[\/latex] but greater than [latex]a[\/latex]. Both [latex]a[\/latex] and [latex]L[\/latex] are real numbers.<\/p>\n<\/div>\n<h2>Understanding Two-Sided Limits<\/h2>\n<p>In the previous example, the left-hand limit and right-hand limit as [latex]x[\/latex] approaches [latex]a[\/latex] are equal. If the left- and right-hand limits are equal, we say that the function [latex]f\\left(x\\right)[\/latex] has a <strong>two-sided limit<\/strong> as [latex]x[\/latex] approaches [latex]a[\/latex]. More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Two-Sided Limit of Function as <em>x<\/em> Approaches <em>a<\/em><\/h3>\n<p>The limit of a function [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]a[\/latex], is equal to [latex]L[\/latex], that is,<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex]<\/p>\n<p>if and only if<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex].<\/p>\n<p>In other words, the left-hand limit of a function [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is equal to the right-hand limit of the same function as [latex]x[\/latex] approaches [latex]a[\/latex]. If such a limit exists, we refer to the limit as a two-sided limit. Otherwise we say the limit does not exist.<\/p>\n<\/div>\n<h2>Finding a Limit Using a Graph<\/h2>\n<p>To visually determine if a limit exists as [latex]x[\/latex] approaches [latex]a[\/latex], we observe the graph of the function when [latex]x[\/latex] is very near to [latex]x=a[\/latex]. In Figure 5&nbsp;we observe the behavior of the graph on both sides of [latex]a[\/latex].<\/p>\n<div style=\"width: 497px\" 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\/27185228\/CNX_Precalc_Figure_12_01_004F2.jpg\" alt=\"Graph of a function that explains the behavior of a limit at (a, L) where the function is increasing when x is less than a and decreasing when x is greater than a.\" width=\"487\" height=\"408\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<p>To determine if a left-hand limit exists, we observe the branch of the graph to the left of [latex]x=a[\/latex], but near [latex]x=a[\/latex]. This is where [latex]x<a[\/latex]. We see that the outputs are getting close to some real number [latex]L[\/latex] so there is a left-hand limit.\n\nTo determine if a right-hand limit exists, observe the branch of the graph to the right of [latex]x=a[\/latex], but near [latex]x=a[\/latex]. This is where [latex]x>a[\/latex]. We see that the outputs are getting close to some real number [latex]L[\/latex], so there is a right-hand limit.<\/p>\n<p>If the left-hand limit and the right-hand limit are the same, as they are in Figure 5, then we know that the function has a two-sided limit. Normally, when we refer to a &#8220;limit,&#8221; we mean a two-sided limit, unless we call it a one-sided limit.<\/p>\n<p>Finally, we can look for an output value for the function [latex]f\\left(x\\right)[\/latex] when the input value [latex]x[\/latex] is equal to [latex]a[\/latex]. The coordinate pair of the point would be [latex]\\left(a,f\\left(a\\right)\\right)[\/latex]. If such a point exists, then [latex]f\\left(a\\right)[\/latex] has a value. If the point does not exist, as in Figure 5, then we say that [latex]f\\left(a\\right)[\/latex] does not exist.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a function [latex]f\\left(x\\right)[\/latex], use a graph to find the limits and a function value as [latex]x[\/latex] approaches [latex]a[\/latex].<\/h3>\n<ol>\n<li>Examine the graph to determine whether a left-hand limit exists.<\/li>\n<li>Examine the graph to determine whether a right-hand limit exists.<\/li>\n<li>If the two one-sided limits exist and are equal, then there is a two-sided limit\u2014what we normally call a &#8220;limit.&#8221;<\/li>\n<li>If there is a point at [latex]x=a[\/latex], then [latex]f\\left(a\\right)[\/latex] is the corresponding function value.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Finding a Limit Using a Graph<\/h3>\n<ol>\n<li>Determine the following limits and function value for the function [latex]f[\/latex] shown in Figure 6.\n<ol>\n<li>[latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n<li>[latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n<li>[latex]f\\left(2\\right)[\/latex]<\/li>\n<\/ol>\n<div style=\"width: 497px\" 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\/27185230\/CNX_Precalc_Figure_12_01_0052.jpg\" alt=\"Graph of a piecewise function that has a positive parabola centered at the origin and goes from negative infinity to (2, 8), an open point, and a decreasing line from (2, 3), a closed point, to positive infinity on the x-axis.\" width=\"487\" height=\"520\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<\/li>\n<li>Determine the following limits and function value for the function [latex]f[\/latex] shown in Figure 7.\n<ol>\n<li>[latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n<li>[latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n<li>[latex]f\\left(2\\right)[\/latex]<\/li>\n<\/ol>\n<div style=\"width: 497px\" 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\/27185232\/CNX_Precalc_Figure_12_01_0062.jpg\" alt=\"Graph of a piecewise function that has a positive parabola from negative infinity to 2 on the x-axis, a decreasing line from 2 to positive infinity on the x-axis, and a point at (2, 4).\" width=\"487\" height=\"483\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q459090\">Show Solution<\/span><\/p>\n<div id=\"q459090\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Looking at Figure 6:\n<ol>\n<li>[latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]; when [latex]x<2[\/latex], but infinitesimally close to 2, the output values get close to [latex]y=8[\/latex].<\/li>\n<li>[latex]\\underset{x\\to 2{}^{+}}{\\mathrm{lim}}f\\left(x\\right)=3[\/latex]; when [latex]x>2[\/latex], but infinitesimally close to 2, the output values approach [latex]y=3[\/latex].<\/li>\n<li>[latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex] does not exist because [latex]\\underset{x\\to 2{}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to 2{}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]; the left and right-hand limits are not equal.<\/li>\n<li>[latex]f\\left(2\\right)=3[\/latex] because the graph of the function [latex]f[\/latex] passes through the point [latex]\\left(2,f\\left(2\\right)\\right)[\/latex] or [latex]\\left(2,3\\right)[\/latex].<\/li>\n<\/ol>\n<\/li>\n<li>Looking at Figure 7:\n<ol>\n<li>[latex]\\underset{x\\to 2{}^{-}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]; when [latex]x<2[\/latex] but infinitesimally close to 2, the output values approach [latex]y=8[\/latex].<\/li>\n<li>[latex]\\underset{x\\to 2{}^{+}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]; when [latex]x>2[\/latex] but infinitesimally close to 2, the output values approach [latex]y=8[\/latex].<\/li>\n<li>[latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex] because [latex]\\underset{x\\to 2{}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to 2{}^{+}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]; the left and right-hand limits are equal.<\/li>\n<li>[latex]f\\left(2\\right)=4[\/latex] because the graph of the function [latex]f[\/latex] passes through the point [latex]\\left(2,f\\left(2\\right)\\right)[\/latex] or [latex]\\left(2,4\\right)[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Using the graph of the function [latex]y=f\\left(x\\right)[\/latex] shown in Figure 8, estimate the following limits.<\/p>\n<div style=\"width: 497px\" 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\/27185234\/CNX_Precalc_Figure_12_01_007n2.jpg\" alt=\"Graph of a piecewise function that has three segments: 1) negative infinity to 0, 2) 0 to 2, and 3) 2 to positive inifnity, which has a discontinuity at (4, 4)\" width=\"487\" height=\"556\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q319433\">Show Solution<\/span><\/p>\n<div id=\"q319433\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. 0; b. 2; c. does not exist; d. [latex]-2[\/latex]; e. 0; f. does not exist; g. 4; h. 4; i. 4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174077\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174077&theme=oea&iframe_resize_id=ohm174077\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Finding a Limit Using a Table<\/h2>\n<p>Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of [latex]x[\/latex] approach [latex]a[\/latex] from both sides. Then we determine if the output values get closer and closer to some real value, the <strong>limit<\/strong> [latex]L[\/latex].<\/p>\n<p>Let\u2019s consider an example using the following function:<\/p>\n<div style=\"text-align: center;\">[latex]\\underset{x\\to 5}{\\mathrm{lim}}\\left(\\dfrac{{x}^{3}-125}{x - 5}\\right)[\/latex]<\/div>\n<p>To create the table, we evaluate the function at values close to [latex]x=5[\/latex]. We use some input values less than 5 and some values greater than 5 as in Figure 9. The table values show that when [latex]x>5[\/latex] but nearing 5, the corresponding output gets close to 75. When [latex]x>5[\/latex] but nearing 5, the corresponding output also gets close to 75.<\/p>\n<div style=\"width: 985px\" 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\/27185237\/CNX_Precalc_Figure_12_01_008n2.jpg\" alt=\"Table shows that as x values approach 5 from the positive or negative direction, f(x) gets very close to 75. But when x is equal to 5, y is undefined.\" width=\"975\" height=\"137\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 9<\/b><\/p>\n<\/div>\n<p>Because<\/p>\n<div style=\"text-align: center;\">[latex]\\underset{x\\to {5}^{-}}{\\mathrm{lim}}f\\left(x\\right)=75=\\underset{x\\to {5}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex],<\/div>\n<p>then<\/p>\n<div style=\"text-align: center;\">[latex]\\underset{x\\to 5}{\\mathrm{lim}}f\\left(x\\right)=75[\/latex].<\/div>\n<p>Remember that [latex]f\\left(5\\right)[\/latex] does not exist.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a function [latex]f[\/latex], use a table to find the limit as [latex]x[\/latex] approaches [latex]a[\/latex] and the value of [latex]f\\left(a\\right)[\/latex], if it exists.<\/h3>\n<ol>\n<li>Choose several input values that approach [latex]a[\/latex] from both the left and right. Record them in a table.<\/li>\n<li>Evaluate the function at each input value. Record them in the table.<\/li>\n<li>Determine if the table values indicate a left-hand limit and a right-hand limit.<\/li>\n<li>If the left-hand and right-hand limits exist and are equal, there is a two-sided limit.<\/li>\n<li>Replace [latex]x[\/latex] with [latex]a[\/latex] to find the value of [latex]f\\left(a\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Finding a Limit Using a Table<\/h3>\n<p>Numerically estimate the limit of the following expression by setting up a table of values on both sides of the limit.<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{5\\sin \\left(x\\right)}{3x}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q431562\">Show Solution<\/span><\/p>\n<div id=\"q431562\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can estimate the value of a limit, if it exists, by evaluating the function at values near [latex]x=0[\/latex]. We cannot find a function value for [latex]x=0[\/latex] directly because the result would have a denominator equal to 0, and thus would be undefined.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{5\\sin \\left(x\\right)}{3x}[\/latex]<\/p>\n<p>We create Figure 10&nbsp;by choosing several input values close to [latex]x=0[\/latex], with half of them less than [latex]x=0[\/latex] and half of them greater than [latex]x=0[\/latex]. Note that we need to be sure we are using radian mode. We evaluate the function at each input value to complete the table.<\/p>\n<p>The table values indicate that when [latex]x<0[\/latex] but approaching 0, the corresponding output nears [latex]\\frac{5}{3}[\/latex].\n\nWhen [latex]x>0[\/latex] but approaching 0, the corresponding output also nears [latex]\\frac{5}{3}[\/latex].<\/p>\n<div style=\"width: 985px\" 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\/27185239\/CNX_Precalc_Figure_12_01_0092.jpg\" alt=\"Table shows that as x values approach 0 from the positive or negative direction, f(x) gets very close to 5 over 3. But when x is equal to 0, y is undefined.\" width=\"975\" height=\"146\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<p>Because<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to {0}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\frac{5}{3}=\\underset{x\\to {0}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex],<\/p>\n<p>then<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\mathrm{lim}}f\\left(x\\right)=\\frac{5}{3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>Is it possible to check our answer using a graphing utility?<\/h3>\n<p><em>Yes. We previously used a table to find a limit of 75 for the function [latex]f\\left(x\\right)=\\frac{{x}^{3}-125}{x - 5}[\/latex] as [latex]x[\/latex] approaches 5. To check, we graph the function on a viewing window as shown in Figure 11. A graphical check shows both branches of the graph of the function get close to the output 75 as [latex]x[\/latex] nears 5. Furthermore, we can use the \u2018trace\u2019 feature of a graphing calculator. By appraoching [latex]x=5[\/latex] we may numerically observe the corresponding outputs getting close to [latex]75[\/latex]. <\/em><\/p>\n<div style=\"width: 497px\" 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\/27185241\/CNX_Precalc_Figure_12_01_010F2.jpg\" alt=\"Graph of an increasing function with a discontinuity at (5, 75)\" width=\"487\" height=\"254\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 11<\/b><\/p>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Numerically estimate the limit of the following function by making a table:<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{20\\sin \\left(x\\right)}{4x}\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q172208\">Show Solution<\/span><\/p>\n<div id=\"q172208\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{20\\sin \\left(x\\right)}{4x}\\right)=5[\/latex]<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185250\/CNX_Precalc_Figure_12_01_0142.jpg\" alt=\"Table showing that f(x) approaches 5 from either side as x approaches 0 from either side.\" \/><\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174079\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174079&theme=oea&iframe_resize_id=ohm174079\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>Is one method for determining a limit better than the other?<\/h3>\n<p><em>No. Both methods have advantages. Graphing allows for quick inspection. Tables can be used when graphical utilities aren\u2019t available, and they can be calculated to a higher precision than could be seen with an unaided eye inspecting a graph.<\/em><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Using a Graphing Utility to Determine a Limit<\/h3>\n<p>With the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as [latex]x[\/latex] approaches 0. If the function has a limit as [latex]x[\/latex] approaches 0, state it. If not, discuss why there is no limit.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=3\\sin \\left(\\frac{\\pi }{x}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q171628\">Show Solution<\/span><\/p>\n<div id=\"q171628\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can use a graphing utility to investigate the behavior of the graph close to [latex]x=0[\/latex]. Centering around [latex]x=0[\/latex], we choose two viewing windows such that the second one is zoomed in closer to [latex]x=0[\/latex] than the first one. The result would resemble Figure 12&nbsp;for [latex]\\left[-2,2\\right][\/latex] by [latex]\\left[-3,3\\right][\/latex].<\/p>\n<div style=\"width: 497px\" 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\/27185243\/CNX_Precalc_Figure_12_01_011F2.jpg\" alt=\"Graph of a sinusodial function zoomed in at &#091;-2, 2&#093; by &#091;-3, 3&#093;.\" width=\"487\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 12<\/b><\/p>\n<\/div>\n<p>The result would resemble Figure 13&nbsp;for [latex]\\left[-0.1,0.1\\right][\/latex] by [latex]\\left[-3,3\\right][\/latex].<\/p>\n<div style=\"width: 498px\" 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\/27185245\/CNX_Precalc_Figure_12_01_012F2.jpg\" alt=\"Graph of the same sinusodial function as in the previous image zoomed in at &#091;-0.1, 0.1&#093; by &#091;-3. 3&#093;.\" width=\"488\" height=\"367\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 13<\/b><\/p>\n<\/div>\n<p>Even closer to zero, we are even less able to distinguish any limits.The closer we get to 0, the greater the swings in the output values are. That is not the behavior of a function with either a left-hand limit or a right-hand limit. And if there is no left-hand limit or right-hand limit, there certainly is no limit to the function [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches 0.We write<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to {0}^{-}}{\\mathrm{lim}}\\left(3\\sin \\left(\\frac{\\pi }{x}\\right)\\right)\\text{ does not exist}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to {0}^{+}}{\\mathrm{lim}}\\left(3\\sin \\left(\\frac{\\pi }{x}\\right)\\right)\\text{ does not exist}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(3\\sin \\left(\\frac{\\pi }{x}\\right)\\right)\\text{ does not exist}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Numerically estimate the following limit: [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\sin \\left(\\dfrac{2}{x}\\right)\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q165056\">Show Solution<\/span><\/p>\n<div id=\"q165056\" class=\"hidden-answer\" style=\"display: none\">\n<p>does not exist<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>A function has a limit if the output values approach some value [latex]L[\/latex] as the input values approach some quantity [latex]a[\/latex].<\/li>\n<li>A shorthand notation is used to describe the limit of a function according to the form [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex], which 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 gets close to [latex]L[\/latex].<\/li>\n<li>A function has a left-hand limit if [latex]f\\left(x\\right)[\/latex] approaches [latex]L[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] where [latex]x<a[\/latex]. A function has a right-hand limit if [latex]f\\left(x\\right)[\/latex] approaches [latex]L[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] where [latex]x>a[\/latex].<\/li>\n<li>A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit.<\/li>\n<li>A graph provides a visual method of determining the limit of a function.<\/li>\n<li>If the function has a limit as [latex]x[\/latex] approaches [latex]a[\/latex], the branches of the graph will approach the same [latex]y\\text{-}[\/latex] coordinate near [latex]x=a[\/latex] from the left and the right.<\/li>\n<li>A table can be used to determine if a function has a limit. The table should show input values that approach [latex]a[\/latex] from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit.<\/li>\n<li>A graphing utility can also be used to find a limit.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137935677\" class=\"definition\">\n<dt>left-hand limit<\/dt>\n<dd id=\"fs-id1165137935682\">the limit of values of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches from [latex]a[\/latex] the left, denoted [latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex]. The values of [latex]f\\left(x\\right)[\/latex] can get as close to the limit [latex]L[\/latex] as we like by taking values of [latex]x[\/latex] sufficiently close to [latex]a[\/latex] such that [latex]x<a[\/latex] and [latex]x\\ne a[\/latex]. Both [latex]a[\/latex] and [latex]L[\/latex] are real numbers.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137673875\" class=\"definition\">\n<dt>limit<\/dt>\n<dd id=\"fs-id1165137673880\">when it exists, the value, [latex]L[\/latex], that the output of a function [latex]f\\left(x\\right)[\/latex] approaches as the input [latex]x[\/latex] gets closer and closer to [latex]a[\/latex] but does not equal [latex]a[\/latex]. The value of the output, [latex]f\\left(x\\right)[\/latex], can get as close to [latex]L[\/latex] as we choose to make it by using input values of [latex]x[\/latex] sufficiently near to [latex]x=a[\/latex], but not necessarily at [latex]x=a[\/latex]. Both [latex]a[\/latex] and [latex]L[\/latex] are real numbers, and [latex]L[\/latex] is denoted [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832351\" class=\"definition\">\n<dt>right-hand limit<\/dt>\n<dd id=\"fs-id1165137832357\">the limit of values of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] from the right, denoted [latex]\\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex]. The values of [latex]f\\left(x\\right)[\/latex] can get as close to the limit [latex]L[\/latex] as we like by taking values of [latex]x[\/latex] sufficiently close to [latex]a[\/latex] where [latex]x>a[\/latex], and [latex]x\\ne a[\/latex]. Both [latex]a[\/latex] and [latex]L[\/latex] are real numbers.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137641237\" class=\"definition\">\n<dt>two-sided limit<\/dt>\n<dd id=\"fs-id1165137641242\">the limit of a function [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]a[\/latex], is equal to [latex]L[\/latex], that is, [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=L[\/latex] if and only if [latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/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-1446\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":503070,"menu_order":1,"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\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1446","chapter","type-chapter","status-publish","hentry"],"part":1445,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1446","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/users\/503070"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1446\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/parts\/1445"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1446\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/media?parent=1446"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapter-type?post=1446"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/contributor?post=1446"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/license?post=1446"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}