{"id":1422,"date":"2024-04-17T13:07:00","date_gmt":"2024-04-17T13:07:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1422"},"modified":"2025-08-17T16:18:29","modified_gmt":"2025-08-17T16:18:29","slug":"introduction-to-the-limit-of-a-function-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/introduction-to-the-limit-of-a-function-fresh-take\/","title":{"raw":"Introduction to the Limit of a Function: Fresh Take","rendered":"Introduction to the Limit of a Function: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Define the limit of a function using proper notation and estimate limits from tables and graphs<\/li>\r\n\t<li>Define one-sided limits with examples and explain their relationship to two-sided limits.<\/li>\r\n\t<li>Describe infinite limits using correct notation and define vertical asymptotes<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>The Definition of a Limit<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Intuitive Definition of a Limit:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches [latex]a[\/latex], [latex]f(x)[\/latex] gets arbitrarily close to [latex]L[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Notation: [latex]\\lim_{x \\to a} f(x) = L[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Estimating Limits: a. Using Tables:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Create tables approaching the point from both sides<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Observe if values converge to a single number b. Using Graphs:<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Examine function behavior near the point of interest<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Look for y-value that the function approaches<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Basic Limit Properties:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} x = a[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} c = c[\/latex] (c is a constant)<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Existence of Limits:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">A limit exists if function values converge to a single, real number<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Limits may not exist due to oscillation or divergence<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571596728\">Evaluate [latex]\\underset{x\\to 0}{\\lim}\\dfrac{\\sin x}{x}[\/latex] using a table of functional values.<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572552454\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572552454\"]We have calculated the values of [latex]f(x)=\\dfrac{(\\sin x)}{x}[\/latex] for the values of [latex]x[\/latex] listed in the table below.<\/p>\r\n<table id=\"fs-id1170572208852\" summary=\"There are two tables. They both have two columns and five rows. The first table has headers x and sin(x)\/x in the first row. Under x in the first column are the values -0.1, -0.01, -0.001, and -0.0001. Under sin(x)\/x in the second column are values 0.998334166468, 0.999983333417, 0.999999833333, and 0.999999998333. The second table has headers x and sin(x)\/x in the first row. Under x in the first column are the values 0.1, 0.01, 0.001, and 0.0001. Under sin(x)\/x in the second column are values 0.998334166468, 0.999983333417, 0.999999833333, and 0.999999998333.\">\r\n<caption>Table of Functional Values for [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}[\/latex]<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]\\frac{\\sin x}{x}[\/latex]<\/th>\r\n<th>\u00a0<\/th>\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]\\frac{\\sin x}{x}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>\u22120.1<\/td>\r\n<td>0.998334166468<\/td>\r\n<td rowspan=\"4\">\u00a0<\/td>\r\n<td>0.1<\/td>\r\n<td>0.998334166468<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.01<\/td>\r\n<td>0.999983333417<\/td>\r\n<td>0.01<\/td>\r\n<td>0.999983333417<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.001<\/td>\r\n<td>0.999999833333<\/td>\r\n<td>0.001<\/td>\r\n<td>0.999999833333<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\u22120.0001<\/td>\r\n<td>0.999999998333<\/td>\r\n<td>0.0001<\/td>\r\n<td>0.999999998333<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170572236179\"><em>Note<\/em>: The values in this table were obtained using a calculator and using all the places given in the calculator output.<\/p>\r\n<p id=\"fs-id1170572558630\">As we read down each [latex]\\frac{\\sin x}{x}[\/latex] column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}=1[\/latex]. A calculator-or computer-generated graph of [latex]f(x)=\\frac{\\sin x}{x}[\/latex] would be similar to that shown in Figure 2, and it confirms our estimate.<\/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\/11202852\/CNX_Calc_Figure_02_02_003.jpg\" alt=\"A graph of f(x) = sin(x)\/x over the interval [-6, 6]. The curving function has a y intercept at x=0 and x intercepts at y=pi and y=-pi.\" width=\"487\" height=\"312\" \/> Figure 2. The graph of [latex]f(x)=(\\sin x)\/x[\/latex] confirms the estimate from the table.[\/caption]\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571596043\">Estimate [latex]\\underset{x\\to 1}{\\lim}\\dfrac{\\frac{1}{x}-1}{x-1}[\/latex] using a table of functional values. Use a graph to confirm your estimate.<\/p>\r\n<p>[reveal-answer q=\"377622\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"377622\"]<\/p>\r\n<p id=\"fs-id1170571656412\">Use [latex]0.9, 0.99, 0.999, 0.9999, 0.99999[\/latex] and [latex]1.1, 1.01, 1.001, 1.0001, 1.00001[\/latex] as your table values.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572227899\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572227899\"]<\/p>\r\n<p id=\"fs-id1170572227899\">[latex]\\underset{x\\to 1}{\\lim}\\frac{\\frac{1}{x}-1}{x-1}=-1[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571656657\">Use the graph of [latex]h(x)[\/latex] in the figure below to evaluate [latex]\\underset{x\\to 2}{\\lim}h(x)[\/latex], if possible.<\/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\/11202902\/CNX_Calc_Figure_02_02_007.jpg\" alt=\"A graph of the function h(x), which is a parabola graphed over [-2.5, 5]. There is an open circle where the vertex should be at the point (2,-1).\" width=\"487\" height=\"431\" \/> Figure 5.\u00a0 The graph of [latex]h(x)[\/latex] consists of a smooth graph with a single removed point at [latex]x=2[\/latex].[\/caption]\r\n[reveal-answer q=\"806443\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"806443\"]\r\n\r\n<p id=\"fs-id1170571657959\">What [latex]y[\/latex]-value does the function approach as the [latex]x[\/latex]-values approach [latex]2[\/latex]?<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170571593051\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170571593051\"]<\/p>\r\n<p id=\"fs-id1170571593051\">[latex]\\underset{x\\to 2}{\\lim}h(x)=-1[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572455169\">Use a table of functional values to evaluate [latex]\\underset{x\\to 2}{\\lim}\\dfrac{|x^2-4|}{x-2}[\/latex], if possible.<\/p>\r\n<p>[reveal-answer q=\"338855\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"338855\"]<\/p>\r\n<p id=\"fs-id1170572560581\">Use [latex]x[\/latex]-values [latex]1.9, 1.99, 1.999, 1.9999, 1.9999[\/latex] and [latex]2.1, 2.01, 2.001, 2.0001, 2.00001 [\/latex] in your table.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572560593\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572560593\"]<\/p>\r\n<p id=\"fs-id1170572560593\">[latex]\\underset{x\\to 2}{\\lim}\\frac{|x^2-4|}{x-2}[\/latex] does not exist.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>One-Sided and Two-Sided Limits<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">One-Sided Limits:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Left-hand limit: [latex]\\lim_{x \\to a^-} f(x) = L[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Right-hand limit: [latex]\\lim_{x \\to a^+} f(x) = L[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Two-Sided Limits:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Exist only if both one-sided limits exist and are equal<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} f(x) = L[\/latex] if and only if [latex]\\lim_{x \\to a^-} f(x) = \\lim_{x \\to a^+} f(x) = L[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Relationship between One-Sided and Two-Sided Limits:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Two-sided limit exists if and only if both one-sided limits exist and are equal<\/li>\r\n\t<li class=\"whitespace-normal break-words\">If one-sided limits differ, the two-sided limit does not exist<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Evaluating One-Sided Limits:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Use tables of values approaching from left or right<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Examine graphs for behavior as [latex]x[\/latex] approaches [latex]a[\/latex] from each side<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571612132\">Use a table of functional values to estimate the following limits, if possible.<\/p>\r\n<ol id=\"fs-id1170571612135\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>[latex]\\underset{x\\to 2^-}{\\lim}\\dfrac{|x^2-4|}{x-2}[\/latex]<\/li>\r\n\t<li>[latex]\\underset{x\\to 2^+}{\\lim}\\dfrac{|x^2-4|}{x-2}[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"228744\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"228744\"]<\/p>\r\n<ol id=\"fs-id1170572452145\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Use [latex]x[\/latex]-values [latex]1.9, 1.99, 1.999, 1.9999, 1.9999[\/latex] to estimate [latex]\\underset{x\\to 2^-}{\\lim}\\frac{|x^2-4|}{x-2}[\/latex].<\/li>\r\n\t<li>Use [latex]x[\/latex]-values [latex]2.1, 2.01, 2.001, 2.0001, 2.00001[\/latex] to estimate [latex]\\underset{x\\to 2^+}{\\lim}\\frac{|x^2-4|}{x-2}[\/latex].<br \/>\r\n(These tables are available from a previous Checkpoint problem.)<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572306438\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572306438\"]<\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\underset{x\\to 2^-}{\\lim}\\frac{|x^2-4|}{x-2}=-4[\/latex]<\/span><\/li>\r\n\t<li><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">latex]\\underset{x\\to 2^+}{\\lim}\\frac{|x^2-4|}{x-2}=4[\/latex]<\/span><\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Evaluate the one-sided and two-sided limits (if they exist) for the following piecewise function:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex] f(x) = \\begin{cases} x^2 + 1, &amp; \\text{if } x &lt; 2 \\ 3x - 5, &amp; \\text{if } x \\geq 2 \\end{cases} [\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">as [latex]x[\/latex] approaches [latex]2[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"326221\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"326221\"]<\/p>\r\n<p>Evaluate the left-hand limit: [<\/p>\r\n<p style=\"text-align: center;\">latex] \\begin{array}{rcl} \\lim_{x \\to 2^-} f(x) &amp;=&amp; \\lim_{x \\to 2^-} (x^2 + 1) \\ &amp;=&amp; 2^2 + 1 \\ &amp;=&amp; 5 \\end{array} [\/latex]<\/p>\r\n<p>Evaluate the right-hand limit:<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rcl} \\lim_{x \\to 2^+} f(x) &amp;=&amp; \\lim_{x \\to 2^+} (3x - 5) \\ &amp;=&amp; 3(2) - 5 \\ &amp;=&amp; 1 \\end{array} [\/latex]<\/p>\r\n<p>Since the one-sided limits are not equal ([latex]5 \u2260 1[\/latex]), the two-sided limit does not exist.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Infinite Limits<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Infinite Limits:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Occur when function values grow without bound<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Notation: [latex]\\lim_{x \\to a} f(x) = \\pm\\infty[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Infinite limits often indicate discontinuities in functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">One-Sided Infinite Limits:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Left-hand: [latex]\\lim_{x \\to a^-} f(x) = \\pm\\infty[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Right-hand: [latex]\\lim_{x \\to a^+} f(x) = \\pm\\infty[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Vertical Asymptotes:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Occur where function approaches infinity as x approaches a value<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Line [latex]x = a[\/latex] is a vertical asymptote if any of: [latex]\\lim_{x \\to a^-} f(x) = \\pm\\infty[\/latex] [latex]\\lim_{x \\to a^+} f(x) = \\pm\\infty[\/latex] [latex]\\lim_{x \\to a} f(x) = \\pm\\infty[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Behavior of [latex]\\frac{1}{(x-a)^n}[\/latex]:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Even [latex]n[\/latex]: [latex]\\lim_{x \\to a} \\frac{1}{(x-a)^n} = +\\infty[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Odd [latex]n[\/latex]: [latex]\\lim_{x \\to a^+} \\frac{1}{(x-a)^n} = +\\infty[\/latex] and [latex]\\lim_{x \\to a^-} \\frac{1}{(x-a)^n} = -\\infty[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571596338\">Evaluate each of the following limits, if possible. Use a table of functional values and graph [latex]f(x)=\\dfrac{1}{x^2}[\/latex] to confirm your conclusion.<\/p>\r\n<ol id=\"fs-id1170571612847\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>[latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x^2}[\/latex]<\/li>\r\n\t<li>[latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x^2}[\/latex]<\/li>\r\n\t<li>[latex]\\underset{x\\to 0}{\\lim}\\frac{1}{x^2}[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"273990\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"273990\"]<\/p>\r\n<p id=\"fs-id1170571612943\">Follow the procedures from the example above.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170571612954\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170571612954\"]<\/p>\r\n<p id=\"fs-id1170571612954\">a. [latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x^2}=+\\infty[\/latex];<\/p>\r\n<p>b. [latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x^2}=+\\infty[\/latex];<\/p>\r\n<p>c. [latex]\\underset{x\\to 0}{\\lim}\\frac{1}{x^2}=+\\infty [\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170571652140\">Evaluate each of the following limits. Identify any vertical asymptotes of the function [latex]f(x)=\\dfrac{1}{(x-2)^3}[\/latex].<\/p>\r\n<ol id=\"fs-id1170571652179\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>[latex]\\underset{x\\to 2^-}{\\lim}\\dfrac{1}{(x-2)^3}[\/latex]<\/li>\r\n\t<li>[latex]\\underset{x\\to 2^+}{\\lim}\\dfrac{1}{(x-2)^3}[\/latex]<\/li>\r\n\t<li>[latex]\\underset{x\\to 2}{\\lim}\\dfrac{1}{(x-2)^3}[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"8356701\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"8356701\"]<\/p>\r\n<p id=\"fs-id1170571545540\">Use the limits summarized under Figure 9.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170571545551\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170571545551\"]<\/p>\r\n<p id=\"fs-id1170571545551\">a. [latex]\\underset{x\\to 2^-}{\\lim}\\frac{1}{(x-2)^3}=\u2212\\infty[\/latex];<\/p>\r\n<p>b. [latex]\\underset{x\\to 2^+}{\\lim}\\frac{1}{(x-2)^3}=+\\infty[\/latex];<\/p>\r\n<p>c. [latex]\\underset{x\\to 2}{\\lim}\\frac{1}{(x-2)^3}[\/latex] DNE. The line [latex]x=2[\/latex] is the vertical asymptote of [latex]f(x)=\\frac{1}{(x-2)^3}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572624475\">Evaluate [latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex] for [latex]f(x)[\/latex] shown here:<\/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\/11202920\/CNX_Calc_Figure_02_02_016.jpg\" alt=\"A graph of a piecewise function. The first segment curves from the third quadrant to the first, crossing through the second quadrant. Where the endpoint would be in the first quadrant is an open circle. The second segment starts at a closed circle a few units below the open circle. It curves down from quadrant one to quadrant four.\" width=\"487\" height=\"350\" \/> Figure 11. Graph of a piecewise function[\/caption]\r\n\r\n<p>[reveal-answer q=\"834551\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"834551\"]<\/p>\r\n<p id=\"fs-id1170572624532\">Compare the limit from the right with the limit from the left.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572624538\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572624538\"]<\/p>\r\n<p id=\"fs-id1170572624538\">Does not exist (DNE).<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Define the limit of a function using proper notation and estimate limits from tables and graphs<\/li>\n<li>Define one-sided limits with examples and explain their relationship to two-sided limits.<\/li>\n<li>Describe infinite limits using correct notation and define vertical asymptotes<\/li>\n<\/ul>\n<\/section>\n<h2>The Definition of a Limit<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Intuitive Definition of a Limit:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches [latex]a[\/latex], [latex]f(x)[\/latex] gets arbitrarily close to [latex]L[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Notation: [latex]\\lim_{x \\to a} f(x) = L[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Estimating Limits: a. Using Tables:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Create tables approaching the point from both sides<\/li>\n<li class=\"whitespace-normal break-words\">Observe if values converge to a single number b. Using Graphs:<\/li>\n<li class=\"whitespace-normal break-words\">Examine function behavior near the point of interest<\/li>\n<li class=\"whitespace-normal break-words\">Look for y-value that the function approaches<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Basic Limit Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} x = a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} c = c[\/latex] (c is a constant)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Existence of Limits:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A limit exists if function values converge to a single, real number<\/li>\n<li class=\"whitespace-normal break-words\">Limits may not exist due to oscillation or divergence<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571596728\">Evaluate [latex]\\underset{x\\to 0}{\\lim}\\dfrac{\\sin x}{x}[\/latex] using a table of functional values.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572552454\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572552454\" class=\"hidden-answer\" style=\"display: none\">We have calculated the values of [latex]f(x)=\\dfrac{(\\sin x)}{x}[\/latex] for the values of [latex]x[\/latex] listed in the table below.<\/p>\n<table id=\"fs-id1170572208852\" summary=\"There are two tables. They both have two columns and five rows. The first table has headers x and sin(x)\/x in the first row. Under x in the first column are the values -0.1, -0.01, -0.001, and -0.0001. Under sin(x)\/x in the second column are values 0.998334166468, 0.999983333417, 0.999999833333, and 0.999999998333. The second table has headers x and sin(x)\/x in the first row. Under x in the first column are the values 0.1, 0.01, 0.001, and 0.0001. Under sin(x)\/x in the second column are values 0.998334166468, 0.999983333417, 0.999999833333, and 0.999999998333.\">\n<caption>Table of Functional Values for [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}[\/latex]<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\frac{\\sin x}{x}[\/latex]<\/th>\n<th>\u00a0<\/th>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]\\frac{\\sin x}{x}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>\u22120.1<\/td>\n<td>0.998334166468<\/td>\n<td rowspan=\"4\">\u00a0<\/td>\n<td>0.1<\/td>\n<td>0.998334166468<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.01<\/td>\n<td>0.999983333417<\/td>\n<td>0.01<\/td>\n<td>0.999983333417<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.001<\/td>\n<td>0.999999833333<\/td>\n<td>0.001<\/td>\n<td>0.999999833333<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\u22120.0001<\/td>\n<td>0.999999998333<\/td>\n<td>0.0001<\/td>\n<td>0.999999998333<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170572236179\"><em>Note<\/em>: The values in this table were obtained using a calculator and using all the places given in the calculator output.<\/p>\n<p id=\"fs-id1170572558630\">As we read down each [latex]\\frac{\\sin x}{x}[\/latex] column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}=1[\/latex]. A calculator-or computer-generated graph of [latex]f(x)=\\frac{\\sin x}{x}[\/latex] would be similar to that shown in Figure 2, and it confirms our estimate.<\/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\/11202852\/CNX_Calc_Figure_02_02_003.jpg\" alt=\"A graph of f(x) = sin(x)\/x over the interval &#091;-6, 6&#093;. The curving function has a y intercept at x=0 and x intercepts at y=pi and y=-pi.\" width=\"487\" height=\"312\" \/><figcaption class=\"wp-caption-text\">Figure 2. The graph of [latex]f(x)=(\\sin x)\/x[\/latex] confirms the estimate from the table.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571596043\">Estimate [latex]\\underset{x\\to 1}{\\lim}\\dfrac{\\frac{1}{x}-1}{x-1}[\/latex] using a table of functional values. Use a graph to confirm your estimate.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q377622\">Hint<\/button><\/p>\n<div id=\"q377622\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571656412\">Use [latex]0.9, 0.99, 0.999, 0.9999, 0.99999[\/latex] and [latex]1.1, 1.01, 1.001, 1.0001, 1.00001[\/latex] as your table values.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572227899\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572227899\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572227899\">[latex]\\underset{x\\to 1}{\\lim}\\frac{\\frac{1}{x}-1}{x-1}=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571656657\">Use the graph of [latex]h(x)[\/latex] in the figure below to evaluate [latex]\\underset{x\\to 2}{\\lim}h(x)[\/latex], if possible.<\/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\/11202902\/CNX_Calc_Figure_02_02_007.jpg\" alt=\"A graph of the function h(x), which is a parabola graphed over [-2.5, 5]. There is an open circle where the vertex should be at the point (2,-1).\" width=\"487\" height=\"431\" \/><figcaption class=\"wp-caption-text\">Figure 5.\u00a0 The graph of [latex]h(x)[\/latex] consists of a smooth graph with a single removed point at [latex]x=2[\/latex].<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q806443\">Hint<\/button><\/p>\n<div id=\"q806443\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571657959\">What [latex]y[\/latex]-value does the function approach as the [latex]x[\/latex]-values approach [latex]2[\/latex]?<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571593051\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571593051\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571593051\">[latex]\\underset{x\\to 2}{\\lim}h(x)=-1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572455169\">Use a table of functional values to evaluate [latex]\\underset{x\\to 2}{\\lim}\\dfrac{|x^2-4|}{x-2}[\/latex], if possible.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q338855\">Hint<\/button><\/p>\n<div id=\"q338855\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572560581\">Use [latex]x[\/latex]-values [latex]1.9, 1.99, 1.999, 1.9999, 1.9999[\/latex] and [latex]2.1, 2.01, 2.001, 2.0001, 2.00001[\/latex] in your table.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572560593\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572560593\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572560593\">[latex]\\underset{x\\to 2}{\\lim}\\frac{|x^2-4|}{x-2}[\/latex] does not exist.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>One-Sided and Two-Sided Limits<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">One-Sided Limits:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Left-hand limit: [latex]\\lim_{x \\to a^-} f(x) = L[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Right-hand limit: [latex]\\lim_{x \\to a^+} f(x) = L[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Two-Sided Limits:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Exist only if both one-sided limits exist and are equal<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\lim_{x \\to a} f(x) = L[\/latex] if and only if [latex]\\lim_{x \\to a^-} f(x) = \\lim_{x \\to a^+} f(x) = L[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relationship between One-Sided and Two-Sided Limits:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Two-sided limit exists if and only if both one-sided limits exist and are equal<\/li>\n<li class=\"whitespace-normal break-words\">If one-sided limits differ, the two-sided limit does not exist<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Evaluating One-Sided Limits:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use tables of values approaching from left or right<\/li>\n<li class=\"whitespace-normal break-words\">Examine graphs for behavior as [latex]x[\/latex] approaches [latex]a[\/latex] from each side<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571612132\">Use a table of functional values to estimate the following limits, if possible.<\/p>\n<ol id=\"fs-id1170571612135\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to 2^-}{\\lim}\\dfrac{|x^2-4|}{x-2}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 2^+}{\\lim}\\dfrac{|x^2-4|}{x-2}[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q228744\">Hint<\/button><\/p>\n<div id=\"q228744\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572452145\" style=\"list-style-type: lower-alpha;\">\n<li>Use [latex]x[\/latex]-values [latex]1.9, 1.99, 1.999, 1.9999, 1.9999[\/latex] to estimate [latex]\\underset{x\\to 2^-}{\\lim}\\frac{|x^2-4|}{x-2}[\/latex].<\/li>\n<li>Use [latex]x[\/latex]-values [latex]2.1, 2.01, 2.001, 2.0001, 2.00001[\/latex] to estimate [latex]\\underset{x\\to 2^+}{\\lim}\\frac{|x^2-4|}{x-2}[\/latex].<br \/>\n(These tables are available from a previous Checkpoint problem.)<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572306438\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572306438\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\underset{x\\to 2^-}{\\lim}\\frac{|x^2-4|}{x-2}=-4[\/latex]<\/span><\/li>\n<li><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">latex]\\underset{x\\to 2^+}{\\lim}\\frac{|x^2-4|}{x-2}=4[\/latex]<\/span><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Evaluate the one-sided and two-sided limits (if they exist) for the following piecewise function:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f(x) = \\begin{cases} x^2 + 1, & \\text{if } x < 2 \\ 3x - 5, & \\text{if } x \\geq 2 \\end{cases}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">as [latex]x[\/latex] approaches [latex]2[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q326221\">Show Answer<\/button><\/p>\n<div id=\"q326221\" class=\"hidden-answer\" style=\"display: none\">\n<p>Evaluate the left-hand limit: [<\/p>\n<p style=\"text-align: center;\">latex] \\begin{array}{rcl} \\lim_{x \\to 2^-} f(x) &amp;=&amp; \\lim_{x \\to 2^-} (x^2 + 1) \\ &amp;=&amp; 2^2 + 1 \\ &amp;=&amp; 5 \\end{array} [\/latex]<\/p>\n<p>Evaluate the right-hand limit:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} \\lim_{x \\to 2^+} f(x) &=& \\lim_{x \\to 2^+} (3x - 5) \\ &=& 3(2) - 5 \\ &=& 1 \\end{array}[\/latex]<\/p>\n<p>Since the one-sided limits are not equal ([latex]5 \u2260 1[\/latex]), the two-sided limit does not exist.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Infinite Limits<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Infinite Limits:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Occur when function values grow without bound<\/li>\n<li class=\"whitespace-normal break-words\">Notation: [latex]\\lim_{x \\to a} f(x) = \\pm\\infty[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Infinite limits often indicate discontinuities in functions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">One-Sided Infinite Limits:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Left-hand: [latex]\\lim_{x \\to a^-} f(x) = \\pm\\infty[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Right-hand: [latex]\\lim_{x \\to a^+} f(x) = \\pm\\infty[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Asymptotes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Occur where function approaches infinity as x approaches a value<\/li>\n<li class=\"whitespace-normal break-words\">Line [latex]x = a[\/latex] is a vertical asymptote if any of: [latex]\\lim_{x \\to a^-} f(x) = \\pm\\infty[\/latex] [latex]\\lim_{x \\to a^+} f(x) = \\pm\\infty[\/latex] [latex]\\lim_{x \\to a} f(x) = \\pm\\infty[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Behavior of [latex]\\frac{1}{(x-a)^n}[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Even [latex]n[\/latex]: [latex]\\lim_{x \\to a} \\frac{1}{(x-a)^n} = +\\infty[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Odd [latex]n[\/latex]: [latex]\\lim_{x \\to a^+} \\frac{1}{(x-a)^n} = +\\infty[\/latex] and [latex]\\lim_{x \\to a^-} \\frac{1}{(x-a)^n} = -\\infty[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571596338\">Evaluate each of the following limits, if possible. Use a table of functional values and graph [latex]f(x)=\\dfrac{1}{x^2}[\/latex] to confirm your conclusion.<\/p>\n<ol id=\"fs-id1170571612847\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x^2}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x^2}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 0}{\\lim}\\frac{1}{x^2}[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q273990\">Hint<\/button><\/p>\n<div id=\"q273990\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571612943\">Follow the procedures from the example above.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571612954\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571612954\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571612954\">a. [latex]\\underset{x\\to 0^-}{\\lim}\\frac{1}{x^2}=+\\infty[\/latex];<\/p>\n<p>b. [latex]\\underset{x\\to 0^+}{\\lim}\\frac{1}{x^2}=+\\infty[\/latex];<\/p>\n<p>c. [latex]\\underset{x\\to 0}{\\lim}\\frac{1}{x^2}=+\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571652140\">Evaluate each of the following limits. Identify any vertical asymptotes of the function [latex]f(x)=\\dfrac{1}{(x-2)^3}[\/latex].<\/p>\n<ol id=\"fs-id1170571652179\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to 2^-}{\\lim}\\dfrac{1}{(x-2)^3}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 2^+}{\\lim}\\dfrac{1}{(x-2)^3}[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 2}{\\lim}\\dfrac{1}{(x-2)^3}[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q8356701\">Hint<\/button><\/p>\n<div id=\"q8356701\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571545540\">Use the limits summarized under Figure 9.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571545551\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571545551\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571545551\">a. [latex]\\underset{x\\to 2^-}{\\lim}\\frac{1}{(x-2)^3}=\u2212\\infty[\/latex];<\/p>\n<p>b. [latex]\\underset{x\\to 2^+}{\\lim}\\frac{1}{(x-2)^3}=+\\infty[\/latex];<\/p>\n<p>c. [latex]\\underset{x\\to 2}{\\lim}\\frac{1}{(x-2)^3}[\/latex] DNE. The line [latex]x=2[\/latex] is the vertical asymptote of [latex]f(x)=\\frac{1}{(x-2)^3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572624475\">Evaluate [latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex] for [latex]f(x)[\/latex] shown here:<\/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\/11202920\/CNX_Calc_Figure_02_02_016.jpg\" alt=\"A graph of a piecewise function. The first segment curves from the third quadrant to the first, crossing through the second quadrant. Where the endpoint would be in the first quadrant is an open circle. The second segment starts at a closed circle a few units below the open circle. It curves down from quadrant one to quadrant four.\" width=\"487\" height=\"350\" \/><figcaption class=\"wp-caption-text\">Figure 11. Graph of a piecewise function<\/figcaption><\/figure>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q834551\">Hint<\/button><\/p>\n<div id=\"q834551\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572624532\">Compare the limit from the right with the limit from the left.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572624538\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572624538\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572624538\">Does not exist (DNE).<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":16,"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":"fresh_take","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\/1422"}],"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":13,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1422\/revisions"}],"predecessor-version":[{"id":4752,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1422\/revisions\/4752"}],"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\/1422\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1422"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1422"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1422"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1422"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}