{"id":283,"date":"2021-02-04T00:48:11","date_gmt":"2021-02-04T00:48:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=283"},"modified":"2022-03-11T21:54:34","modified_gmt":"2022-03-11T21:54:34","slug":"evaluating-limits","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/evaluating-limits\/","title":{"raw":"Evaluating Limits","rendered":"Evaluating Limits"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Recognize the basic limit laws<\/li>\r\n \t<li>Use the limit laws to evaluate the limit of a function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Limit Laws<\/h2>\r\n<p id=\"fs-id1170571680609\">The first two limit laws were stated earlier in the course and we repeat them here. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions.<\/p>\r\n\r\n<div id=\"fs-id1170572451153\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Basic Limit Results<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170572205248\">For any real number [latex]a[\/latex] and any constant [latex]c[\/latex],<\/p>\r\n\r\n<ol id=\"fs-id1170572286963\">\r\n \t<li>\r\n<div id=\"fs-id1170572624896\" class=\"equation\">[latex]\\underset{x\\to a}{\\lim}x=a[\/latex]<\/div><\/li>\r\n \t<li>\r\n<div id=\"fs-id1170572209025\" class=\"equation\">[latex]\\underset{x\\to a}{\\lim}c=c[\/latex]<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170572111463\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating a Basic Limit<\/h3>\r\n<p id=\"fs-id1170571569246\">Evaluate each of the following limits using the basic limit results above.<\/p>\r\n\r\n<ol id=\"fs-id1170572176731\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\underset{x\\to 2}{\\lim}x[\/latex]<\/li>\r\n \t<li>[latex]\\underset{x\\to 2}{\\lim}5[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170572101621\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572101621\"]\r\n<ol id=\"fs-id1170572101621\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>The limit of [latex]x[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is [latex]a[\/latex]: [latex]\\underset{x\\to 2}{\\lim}x=2[\/latex].<\/li>\r\n \t<li>The limit of a constant is that constant: [latex]\\underset{x\\to 2}{\\lim}5=5[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]4886[\/ohm_question]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572570027\">We now take a look at the limit laws, the individual properties of limits. The proofs that these laws hold are omitted here.<\/p>\r\n\r\n<div id=\"fs-id1170572508800\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Limit Laws<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170572086164\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be defined for all [latex]x\\ne a[\/latex] over some open interval containing [latex]a[\/latex]. Assume that [latex]L[\/latex] and [latex]M[\/latex] are real numbers such that [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex] and [latex]\\underset{x\\to a}{\\lim}g(x)=M[\/latex]. Let [latex]c[\/latex] be a constant. Then, each of the following statements holds:<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1170572204187\"><strong>Sum law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}(f(x)+g(x))=\\underset{x\\to a}{\\lim}f(x)+\\underset{x\\to a}{\\lim}g(x)=L+M[\/latex]<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1170572627273\"><strong>Difference law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}(f(x)-g(x))=\\underset{x\\to a}{\\lim}f(x)-\\underset{x\\to a}{\\lim}g(x)=L-M[\/latex]<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1170572450574\"><strong>Constant multiple law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}cf(x)=c \\cdot \\underset{x\\to a}{\\lim}f(x)=cL[\/latex]<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1170572104032\"><strong>Product law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}(f(x) \\cdot g(x))=\\underset{x\\to a}{\\lim}f(x) \\cdot \\underset{x\\to a}{\\lim}g(x)=L \\cdot M[\/latex]<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1170572347458\"><strong>Quotient law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}\\dfrac{f(x)}{g(x)}=\\dfrac{\\underset{x\\to a}{\\lim}f(x)}{\\underset{x\\to a}{\\lim}g(x)}=\\frac{L}{M}[\/latex] for [latex]M\\ne 0[\/latex]<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1170572246193\"><strong>Power law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}(f(x))^n=(\\underset{x\\to a}{\\lim}f(x))^n=L^n[\/latex] for every positive integer [latex]n[\/latex].<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1170572232633\"><strong>Root law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}\\sqrt[n]{f(x)}=\\sqrt[n]{\\underset{x\\to a}{\\lim}f(x)}=\\sqrt[n]{L}[\/latex] for all [latex]L[\/latex] if [latex]n[\/latex] is odd and for [latex]L\\ge 0[\/latex] if [latex]n[\/latex] is even<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572479215\">We now practice applying these limit laws to evaluate a limit.<\/p>\r\n\r\n<div id=\"fs-id1170572451489\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating a Limit Using Limit Laws<\/h3>\r\n<p id=\"fs-id1170572109838\">Use the limit laws to evaluate [latex]\\underset{x\\to -3}{\\lim}(4x+2)[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170572169042\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572169042\"]\r\n<p id=\"fs-id1170572169042\">Let\u2019s apply the limit laws one step at a time to be sure we understand how they work. We need to keep in mind the requirement that, at each application of a limit law, the new limits must exist for the limit law to be applied.<\/p>\r\n<p id=\"fs-id1170571565987\">[latex]\\begin{array}{ccccc}\\underset{x\\to -3}{\\lim}(4x+2)\\hfill &amp; =\\underset{x\\to -3}{\\lim}4x+\\underset{x\\to -3}{\\lim}2\\hfill &amp; &amp; &amp; \\text{Apply the sum law.}\\hfill \\\\ &amp; =4 \\cdot \\underset{x\\to -3}{\\lim}x+\\underset{x\\to -3}{\\lim}2\\hfill &amp; &amp; &amp; \\text{Apply the constant multiple law.}\\hfill \\\\ &amp; =4 \\cdot (-3)+2=-10\\hfill &amp; &amp; &amp; \\text{Apply the basic limit results and simplify.}\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572509954\" class=\"textbook exercises\">\r\n<h3>Example: Using Limit Laws Repeatedly<\/h3>\r\nUse the limit laws to evaluate [latex]\\underset{x\\to 2}{\\lim}\\dfrac{2x^2-3x+1}{x^3+4}[\/latex].\r\n\r\n[reveal-answer q=\"fs-id1170572506406\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572506406\"]\r\n<p id=\"fs-id1170572506406\">To find this limit, we need to apply the limit laws several times. Again, we need to keep in mind that as we rewrite the limit in terms of other limits, each new limit must exist for the limit law to be applied.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccc}\\\\ \\\\ \\underset{x\\to 2}{\\lim}\\large \\frac{2x^2-3x+1}{x^3+4} &amp; = \\large \\frac{\\underset{x\\to 2}{\\lim}(2x^2-3x+1)}{\\underset{x\\to 2}{\\lim}(x^3+4)} &amp; &amp; &amp; \\text{Apply the quotient law, making sure that} \\, 2^3+4\\ne 0 \\\\ &amp; = \\large \\frac{2 \\cdot \\underset{x\\to 2}{\\lim}x^2-3 \\cdot \\underset{x\\to 2}{\\lim}x+\\underset{x\\to 2}{\\lim}1}{\\underset{x\\to 2}{\\lim}x^3+\\underset{x\\to 2}{\\lim}4} &amp; &amp; &amp; \\text{Apply the sum law and constant multiple law.} \\\\ &amp; = \\large \\frac{2 \\cdot (\\underset{x\\to 2}{\\lim}x)^2-3 \\cdot \\underset{x\\to 2}{\\lim}x+\\underset{x\\to 2}{\\lim}1}{(\\underset{x\\to 2}{\\lim}x)^3+\\underset{x\\to 2}{\\lim}4} &amp; &amp; &amp; \\text{Apply the power law.} \\\\ &amp; = \\large \\frac{2(4)-3(2)+1}{2^3+4}=\\frac{1}{4} &amp; &amp; &amp; \\text{Apply the basic limit laws and simplify.} \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170571655486\">Use the limit laws to evaluate [latex]\\underset{x\\to 6}{\\lim}(2x-1)\\sqrt{x+4}[\/latex]. In each step, indicate the limit law applied.<\/p>\r\n[reveal-answer q=\"6635113\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"6635113\"]\r\n<p id=\"fs-id1170572209920\">Begin by applying the product law.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572094142\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572094142\"]\r\n<p id=\"fs-id1170572094142\">[latex]11\\sqrt{10}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Limits of Polynomial and Rational Functions<\/h2>\r\n<p id=\"fs-id1170572133214\">By now you have probably noticed that, in each of the previous examples, it has been the case that [latex]\\underset{x\\to a}{\\lim}f(x)=f(a)[\/latex]. This is not always true, but it does hold for all polynomials for any choice of [latex]a[\/latex] and for all rational functions at all values of [latex]a[\/latex] for which the rational function is defined.<\/p>\r\n\r\n<div id=\"fs-id1170572557796\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Limits of Polynomial and Rational Functions<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170572557802\">Let [latex]p(x)[\/latex] and [latex]q(x)[\/latex] be polynomial functions. Let [latex]a[\/latex] be a real number. Then,<\/p>\r\n\r\n<div id=\"fs-id1170572347161\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}p(x)=p(a)[\/latex]<\/div>\r\n&nbsp;\r\n<div><\/div>\r\n<div><\/div>\r\n<div id=\"fs-id1170571656084\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}\\dfrac{p(x)}{q(x)}=\\dfrac{p(a)}{q(a)} \\, \\text{when} \\, q(a)\\ne 0[\/latex]<\/div>\r\n<div><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170571650163\">To see that this theorem holds, consider the polynomial [latex]p(x)=c_nx^n+c_{n-1}x^{n-1}+\\cdots +c_1x+c_0[\/latex]. By applying the sum, constant multiple, and power laws, we end up with<\/p>\r\n\r\n<div id=\"fs-id1170571648575\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cc}\\hfill \\underset{x\\to a}{\\lim}p(x)&amp; =\\underset{x\\to a}{\\lim}(c_nx^n+c_{n-1}x^{n-1}+\\cdots +c_1x+c_0)\\hfill \\\\ &amp; =c_n(\\underset{x\\to a}{\\lim}x)^n+c_{n-1}(\\underset{x\\to a}{\\lim}x)^{n-1}+\\cdots +c_1(\\underset{x\\to a}{\\lim}x)+\\underset{x\\to a}{\\lim}c_0\\hfill \\\\ &amp; =c_na^n+c_{n-1}a^{n-1}+\\cdots +c_1a+c_0\\hfill \\\\ &amp; =p(a)\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572628443\">It now follows from the quotient law that if [latex]p(x)[\/latex] and [latex]q(x)[\/latex] are polynomials for which [latex]q(a)\\ne 0[\/latex], then<\/p>\r\n\r\n<div id=\"fs-id1170571672249\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}\\dfrac{p(x)}{q(x)}=\\dfrac{p(a)}{q(a)}[\/latex]<\/div>\r\n<p id=\"fs-id1170572305824\">The example below applies this result.<\/p>\r\n\r\n<div id=\"fs-id1170572305829\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating a Limit of a Rational Function<\/h3>\r\n<p id=\"fs-id1170572305839\">Evaluate the [latex]\\underset{x\\to 3}{\\lim}\\dfrac{2x^2-3x+1}{5x+4}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170572305892\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572305892\"]\r\n<p id=\"fs-id1170572305892\">Since 3 is in the domain of the rational function [latex]f(x)=\\frac{2x^2-3x+1}{5x+4}[\/latex], we can calculate the limit by substituting 3 for [latex]x[\/latex] into the function. Thus,<\/p>\r\n\r\n<div id=\"fs-id1170571686198\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 3}{\\lim}\\dfrac{2x^2-3x+1}{5x+4}=\\dfrac{10}{19}[\/latex]\r\n[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571675270\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170571675277\">Evaluate [latex]\\underset{x\\to -2}{\\lim}(3x^3-2x+7)[\/latex].<\/p>\r\n[reveal-answer q=\"4482011\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"4482011\"]\r\n<p id=\"fs-id1170571688063\">Use\u00a0limits of polynomial and rational functions<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170571688072\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571688072\"]\r\n<p id=\"fs-id1170571688072\">[latex]\u221213[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solutions to all examples and try it's on this page.[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/Jv0Wi-JERjo?controls=0&amp;start=117&amp;end=328&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.3LimitLaws117to328_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.3 Limit Laws\" here (opens in new window)<\/a>.[\/hidden-answer]","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Recognize the basic limit laws<\/li>\n<li>Use the limit laws to evaluate the limit of a function<\/li>\n<\/ul>\n<\/div>\n<h2>Limit Laws<\/h2>\n<p id=\"fs-id1170571680609\">The first two limit laws were stated earlier in the course and we repeat them here. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions.<\/p>\n<div id=\"fs-id1170572451153\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Basic Limit Results<\/h3>\n<hr \/>\n<p id=\"fs-id1170572205248\">For any real number [latex]a[\/latex] and any constant [latex]c[\/latex],<\/p>\n<ol id=\"fs-id1170572286963\">\n<li>\n<div id=\"fs-id1170572624896\" class=\"equation\">[latex]\\underset{x\\to a}{\\lim}x=a[\/latex]<\/div>\n<\/li>\n<li>\n<div id=\"fs-id1170572209025\" class=\"equation\">[latex]\\underset{x\\to a}{\\lim}c=c[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170572111463\" class=\"textbook exercises\">\n<h3>Example: Evaluating a Basic Limit<\/h3>\n<p id=\"fs-id1170571569246\">Evaluate each of the following limits using the basic limit results above.<\/p>\n<ol id=\"fs-id1170572176731\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\underset{x\\to 2}{\\lim}x[\/latex]<\/li>\n<li>[latex]\\underset{x\\to 2}{\\lim}5[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572101621\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572101621\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572101621\" style=\"list-style-type: lower-alpha;\">\n<li>The limit of [latex]x[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is [latex]a[\/latex]: [latex]\\underset{x\\to 2}{\\lim}x=2[\/latex].<\/li>\n<li>The limit of a constant is that constant: [latex]\\underset{x\\to 2}{\\lim}5=5[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm4886\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4886&theme=oea&iframe_resize_id=ohm4886&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p id=\"fs-id1170572570027\">We now take a look at the limit laws, the individual properties of limits. The proofs that these laws hold are omitted here.<\/p>\n<div id=\"fs-id1170572508800\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Limit Laws<\/h3>\n<hr \/>\n<p id=\"fs-id1170572086164\">Let [latex]f(x)[\/latex] and [latex]g(x)[\/latex] be defined for all [latex]x\\ne a[\/latex] over some open interval containing [latex]a[\/latex]. Assume that [latex]L[\/latex] and [latex]M[\/latex] are real numbers such that [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex] and [latex]\\underset{x\\to a}{\\lim}g(x)=M[\/latex]. Let [latex]c[\/latex] be a constant. Then, each of the following statements holds:<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572204187\"><strong>Sum law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}(f(x)+g(x))=\\underset{x\\to a}{\\lim}f(x)+\\underset{x\\to a}{\\lim}g(x)=L+M[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572627273\"><strong>Difference law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}(f(x)-g(x))=\\underset{x\\to a}{\\lim}f(x)-\\underset{x\\to a}{\\lim}g(x)=L-M[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572450574\"><strong>Constant multiple law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}cf(x)=c \\cdot \\underset{x\\to a}{\\lim}f(x)=cL[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572104032\"><strong>Product law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}(f(x) \\cdot g(x))=\\underset{x\\to a}{\\lim}f(x) \\cdot \\underset{x\\to a}{\\lim}g(x)=L \\cdot M[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572347458\"><strong>Quotient law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}\\dfrac{f(x)}{g(x)}=\\dfrac{\\underset{x\\to a}{\\lim}f(x)}{\\underset{x\\to a}{\\lim}g(x)}=\\frac{L}{M}[\/latex] for [latex]M\\ne 0[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572246193\"><strong>Power law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}(f(x))^n=(\\underset{x\\to a}{\\lim}f(x))^n=L^n[\/latex] for every positive integer [latex]n[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572232633\"><strong>Root law for limits<\/strong>: [latex]\\underset{x\\to a}{\\lim}\\sqrt[n]{f(x)}=\\sqrt[n]{\\underset{x\\to a}{\\lim}f(x)}=\\sqrt[n]{L}[\/latex] for all [latex]L[\/latex] if [latex]n[\/latex] is odd and for [latex]L\\ge 0[\/latex] if [latex]n[\/latex] is even<\/p>\n<\/div>\n<p id=\"fs-id1170572479215\">We now practice applying these limit laws to evaluate a limit.<\/p>\n<div id=\"fs-id1170572451489\" class=\"textbook exercises\">\n<h3>Example: Evaluating a Limit Using Limit Laws<\/h3>\n<p id=\"fs-id1170572109838\">Use the limit laws to evaluate [latex]\\underset{x\\to -3}{\\lim}(4x+2)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572169042\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572169042\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572169042\">Let\u2019s apply the limit laws one step at a time to be sure we understand how they work. We need to keep in mind the requirement that, at each application of a limit law, the new limits must exist for the limit law to be applied.<\/p>\n<p id=\"fs-id1170571565987\">[latex]\\begin{array}{ccccc}\\underset{x\\to -3}{\\lim}(4x+2)\\hfill & =\\underset{x\\to -3}{\\lim}4x+\\underset{x\\to -3}{\\lim}2\\hfill & & & \\text{Apply the sum law.}\\hfill \\\\ & =4 \\cdot \\underset{x\\to -3}{\\lim}x+\\underset{x\\to -3}{\\lim}2\\hfill & & & \\text{Apply the constant multiple law.}\\hfill \\\\ & =4 \\cdot (-3)+2=-10\\hfill & & & \\text{Apply the basic limit results and simplify.}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572509954\" class=\"textbook exercises\">\n<h3>Example: Using Limit Laws Repeatedly<\/h3>\n<p>Use the limit laws to evaluate [latex]\\underset{x\\to 2}{\\lim}\\dfrac{2x^2-3x+1}{x^3+4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572506406\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572506406\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572506406\">To find this limit, we need to apply the limit laws several times. Again, we need to keep in mind that as we rewrite the limit in terms of other limits, each new limit must exist for the limit law to be applied.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccc}\\\\ \\\\ \\underset{x\\to 2}{\\lim}\\large \\frac{2x^2-3x+1}{x^3+4} & = \\large \\frac{\\underset{x\\to 2}{\\lim}(2x^2-3x+1)}{\\underset{x\\to 2}{\\lim}(x^3+4)} & & & \\text{Apply the quotient law, making sure that} \\, 2^3+4\\ne 0 \\\\ & = \\large \\frac{2 \\cdot \\underset{x\\to 2}{\\lim}x^2-3 \\cdot \\underset{x\\to 2}{\\lim}x+\\underset{x\\to 2}{\\lim}1}{\\underset{x\\to 2}{\\lim}x^3+\\underset{x\\to 2}{\\lim}4} & & & \\text{Apply the sum law and constant multiple law.} \\\\ & = \\large \\frac{2 \\cdot (\\underset{x\\to 2}{\\lim}x)^2-3 \\cdot \\underset{x\\to 2}{\\lim}x+\\underset{x\\to 2}{\\lim}1}{(\\underset{x\\to 2}{\\lim}x)^3+\\underset{x\\to 2}{\\lim}4} & & & \\text{Apply the power law.} \\\\ & = \\large \\frac{2(4)-3(2)+1}{2^3+4}=\\frac{1}{4} & & & \\text{Apply the basic limit laws and simplify.} \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170571655486\">Use the limit laws to evaluate [latex]\\underset{x\\to 6}{\\lim}(2x-1)\\sqrt{x+4}[\/latex]. In each step, indicate the limit law applied.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q6635113\">Hint<\/span><\/p>\n<div id=\"q6635113\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572209920\">Begin by applying the product law.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572094142\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572094142\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572094142\">[latex]11\\sqrt{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Limits of Polynomial and Rational Functions<\/h2>\n<p id=\"fs-id1170572133214\">By now you have probably noticed that, in each of the previous examples, it has been the case that [latex]\\underset{x\\to a}{\\lim}f(x)=f(a)[\/latex]. This is not always true, but it does hold for all polynomials for any choice of [latex]a[\/latex] and for all rational functions at all values of [latex]a[\/latex] for which the rational function is defined.<\/p>\n<div id=\"fs-id1170572557796\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Limits of Polynomial and Rational Functions<\/h3>\n<hr \/>\n<p id=\"fs-id1170572557802\">Let [latex]p(x)[\/latex] and [latex]q(x)[\/latex] be polynomial functions. Let [latex]a[\/latex] be a real number. Then,<\/p>\n<div id=\"fs-id1170572347161\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}p(x)=p(a)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div><\/div>\n<div><\/div>\n<div id=\"fs-id1170571656084\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}\\dfrac{p(x)}{q(x)}=\\dfrac{p(a)}{q(a)} \\, \\text{when} \\, q(a)\\ne 0[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<p id=\"fs-id1170571650163\">To see that this theorem holds, consider the polynomial [latex]p(x)=c_nx^n+c_{n-1}x^{n-1}+\\cdots +c_1x+c_0[\/latex]. By applying the sum, constant multiple, and power laws, we end up with<\/p>\n<div id=\"fs-id1170571648575\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cc}\\hfill \\underset{x\\to a}{\\lim}p(x)& =\\underset{x\\to a}{\\lim}(c_nx^n+c_{n-1}x^{n-1}+\\cdots +c_1x+c_0)\\hfill \\\\ & =c_n(\\underset{x\\to a}{\\lim}x)^n+c_{n-1}(\\underset{x\\to a}{\\lim}x)^{n-1}+\\cdots +c_1(\\underset{x\\to a}{\\lim}x)+\\underset{x\\to a}{\\lim}c_0\\hfill \\\\ & =c_na^n+c_{n-1}a^{n-1}+\\cdots +c_1a+c_0\\hfill \\\\ & =p(a)\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572628443\">It now follows from the quotient law that if [latex]p(x)[\/latex] and [latex]q(x)[\/latex] are polynomials for which [latex]q(a)\\ne 0[\/latex], then<\/p>\n<div id=\"fs-id1170571672249\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}\\dfrac{p(x)}{q(x)}=\\dfrac{p(a)}{q(a)}[\/latex]<\/div>\n<p id=\"fs-id1170572305824\">The example below applies this result.<\/p>\n<div id=\"fs-id1170572305829\" class=\"textbook exercises\">\n<h3>Example: Evaluating a Limit of a Rational Function<\/h3>\n<p id=\"fs-id1170572305839\">Evaluate the [latex]\\underset{x\\to 3}{\\lim}\\dfrac{2x^2-3x+1}{5x+4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572305892\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572305892\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572305892\">Since 3 is in the domain of the rational function [latex]f(x)=\\frac{2x^2-3x+1}{5x+4}[\/latex], we can calculate the limit by substituting 3 for [latex]x[\/latex] into the function. Thus,<\/p>\n<div id=\"fs-id1170571686198\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to 3}{\\lim}\\dfrac{2x^2-3x+1}{5x+4}=\\dfrac{10}{19}[\/latex]\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571675270\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170571675277\">Evaluate [latex]\\underset{x\\to -2}{\\lim}(3x^3-2x+7)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q4482011\">Hint<\/span><\/p>\n<div id=\"q4482011\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571688063\">Use\u00a0limits of polynomial and rational functions<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571688072\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571688072\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571688072\">[latex]\u221213[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solutions to all examples and try it&#8217;s on this page.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/Jv0Wi-JERjo?controls=0&amp;start=117&amp;end=328&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.3LimitLaws117to328_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.3 Limit Laws&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\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-283\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>2.3 Limit Laws. <strong>Authored by<\/strong>: Ryan Melton. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/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":17533,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"2.3 Limit Laws\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"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-283","chapter","type-chapter","status-publish","hentry"],"part":28,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/283","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":23,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/283\/revisions"}],"predecessor-version":[{"id":4787,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/283\/revisions\/4787"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/28"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/283\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=283"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=283"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=283"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=283"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}