{"id":547,"date":"2025-10-09T15:21:01","date_gmt":"2025-10-09T15:21:01","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=547"},"modified":"2025-11-20T14:59:29","modified_gmt":"2025-11-20T14:59:29","slug":"vector-valued-functions-and-motion-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/vector-valued-functions-and-motion-background-youll-need-1\/","title":{"raw":"Vector-Valued Functions and Motion: Background You'll Need 1","rendered":"Vector-Valued Functions and Motion: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Use basic rules to find limits of functions, including polynomial and rational functions<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>The Limit Laws<\/h2>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Limit laws<\/strong> are algebraic rules that let us break down complicated limits into simpler parts. Instead of evaluating limits from scratch each time, we can use these properties to handle complex functions systematically.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>basic limit properties<\/h3>\r\n<p id=\"fs-id1170572205248\">Let [latex]a[\/latex] be a real number and [latex]c[\/latex] be a constant.<\/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<\/section>\r\n<div id=\"fs-id1170572111463\" class=\"textbook exercises\"><section class=\"textbox example\">\r\n<div id=\"fs-id1170572111463\" class=\"textbook exercises\">\r\n<p id=\"fs-id1170571569246\">Evaluate each of the following limits using the basic limit properties 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<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]288275[\/ohm_question]<\/section><\/div>\r\nThe limit laws outline the essential properties of limits, crucial for the systematic evaluation of functions as they approach specific points.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>the limit laws<\/h3>\r\n<p id=\"fs-id1170572086164\">For all [latex]x[\/latex] near [latex]a[\/latex], consider functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex] with limits [latex]L[\/latex] and [latex]M [\/latex] respectively, [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. The following are established limit laws:<\/p>\r\n&nbsp;\r\n\r\n<strong>Sum law for limits<\/strong>:\r\n\r\n<center>[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]<\/center>&nbsp;\r\n\r\n<strong>Difference law for limits<\/strong>:\r\n\r\n<center>[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]<\/center>&nbsp;\r\n\r\n<strong>Constant multiple law for limits<\/strong>:\r\n\r\n<center>[latex]\\underset{x\\to a}{\\lim}cf(x)=c \\cdot \\underset{x\\to a}{\\lim}f(x)=cL[\/latex]<\/center>&nbsp;\r\n\r\n<strong>Product law for limits<\/strong>:\r\n\r\n<center>[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]<\/center>&nbsp;\r\n\r\n<strong>Quotient law for limits<\/strong>:\r\n\r\n<center>[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]<\/center>&nbsp;\r\n\r\n<strong>Power law for limits<\/strong>:\r\n\r\n<center>[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]<\/center>\r\n<p id=\"fs-id1170572246193\">.<\/p>\r\n&nbsp;\r\n\r\n<strong>Root law for limits<\/strong>:\r\n\r\n<center>[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<\/center><\/section><section class=\"textbox proTip\">A handy way to keep the limit laws top of mind is to associate them with simple arithmetic operations you already know:\r\n<ul>\r\n \t<li><strong>Sum and Difference<\/strong>: Just like adding or subtracting numbers, you can add or subtract limits.<\/li>\r\n \t<li><strong>Constant Multiples<\/strong>: Multiplying a number by a constant? The same goes for a limit.<\/li>\r\n \t<li><strong>Product<\/strong>: Multiplying two numbers? You can also multiply their limits.<\/li>\r\n \t<li><strong>Quotient<\/strong>: Dividing numbers translates to dividing their limits, just watch out for a zero denominator.<\/li>\r\n \t<li><strong>Powers and Roots<\/strong>: Raising a number to a power or taking a root? Apply the same operation to the limit.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\">\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\" style=\"text-align: center;\">[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<\/section><section class=\"textbox example\">Use the limit laws to evaluate [latex]\\underset{x\\to 2}{\\lim}\\dfrac{2x^2-3x+1}{x^3+4}[\/latex].[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<\/section>\r\n<div id=\"fs-id1170571675270\" class=\"textbook key-takeaways\">\r\n<h3>Limits of Polynomial and Rational Functions<\/h3>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">When working with polynomial and rational functions, you can often find limits by direct substitution.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Consider the limit [latex]\\underset{x\\to 2}{\\text{lim}}\\dfrac{2x^2-3x+1}{x^3+4}[\/latex]. Using limit laws step-by-step gives us [latex]\\frac{1}{4}[\/latex]. But notice what happens if we simply substitute [latex]x = 2[\/latex] directly into the function:<\/p>\r\n<p style=\"text-align: center;\">[latex]f(2) = \\dfrac{2(2)^2-3(2)+1}{(2)^3+4} = \\dfrac{8-6+1}{8+4} = \\dfrac{3}{12} = \\frac{1}{4}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">We get the same answer.<\/p>\r\n<p id=\"fs-id1170572133214\">This is not mere chance. It demonstrates a foundational concept in calculus: for polynomial and rational functions that are continuous at a point [latex]a[\/latex], the limit as [latex]x[\/latex] approaches [latex]a[\/latex] equals the value of the function at [latex]a[\/latex], or [latex]f(a)[\/latex]. This holds true provided the function is defined at that point.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>limits of polynomial and rational functions<\/h3>\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<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<\/section><section class=\"textbox questionHelp\">\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].<\/p>\r\nBy applying the sum, constant multiple, and power laws, we end up with:\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<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<\/section><section class=\"textbox example\">\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 [latex]3[\/latex] 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 [latex]3[\/latex] for [latex]x[\/latex] into the function.<\/p>\r\n<p style=\"text-align: center;\">[latex]f(3)=\\frac{2(3)^2-3(3)+1}{5(3)+4} = \\dfrac{10}{19}[\/latex]<\/p>\r\nThus,\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<\/section>\r\n<div><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]288276[\/ohm_question]<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Use basic rules to find limits of functions, including polynomial and rational functions<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>The Limit Laws<\/h2>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Limit laws<\/strong> are algebraic rules that let us break down complicated limits into simpler parts. Instead of evaluating limits from scratch each time, we can use these properties to handle complex functions systematically.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>basic limit properties<\/h3>\n<p id=\"fs-id1170572205248\">Let [latex]a[\/latex] be a real number and [latex]c[\/latex] be a constant.<\/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<\/section>\n<div id=\"fs-id1170572111463\" class=\"textbook exercises\">\n<section class=\"textbox example\">\n<div class=\"textbook exercises\">\n<p id=\"fs-id1170571569246\">Evaluate each of the following limits using the basic limit properties 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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572101621\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm288275\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288275&theme=lumen&iframe_resize_id=ohm288275&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<p>The limit laws outline the essential properties of limits, crucial for the systematic evaluation of functions as they approach specific points.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>the limit laws<\/h3>\n<p id=\"fs-id1170572086164\">For all [latex]x[\/latex] near [latex]a[\/latex], consider functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex] with limits [latex]L[\/latex] and [latex]M[\/latex] respectively, [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. The following are established limit laws:<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Sum law for limits<\/strong>:<\/p>\n<div style=\"text-align: center;\">[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]<\/div>\n<p>&nbsp;<\/p>\n<p><strong>Difference law for limits<\/strong>:<\/p>\n<div style=\"text-align: center;\">[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]<\/div>\n<p>&nbsp;<\/p>\n<p><strong>Constant multiple law for limits<\/strong>:<\/p>\n<div style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}cf(x)=c \\cdot \\underset{x\\to a}{\\lim}f(x)=cL[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p><strong>Product law for limits<\/strong>:<\/p>\n<div style=\"text-align: center;\">[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]<\/div>\n<p>&nbsp;<\/p>\n<p><strong>Quotient law for limits<\/strong>:<\/p>\n<div style=\"text-align: center;\">[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]<\/div>\n<p>&nbsp;<\/p>\n<p><strong>Power law for limits<\/strong>:<\/p>\n<div style=\"text-align: center;\">[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]<\/div>\n<p id=\"fs-id1170572246193\">.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Root law for limits<\/strong>:<\/p>\n<div style=\"text-align: center;\">[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<\/div>\n<\/section>\n<section class=\"textbox proTip\">A handy way to keep the limit laws top of mind is to associate them with simple arithmetic operations you already know:<\/p>\n<ul>\n<li><strong>Sum and Difference<\/strong>: Just like adding or subtracting numbers, you can add or subtract limits.<\/li>\n<li><strong>Constant Multiples<\/strong>: Multiplying a number by a constant? The same goes for a limit.<\/li>\n<li><strong>Product<\/strong>: Multiplying two numbers? You can also multiply their limits.<\/li>\n<li><strong>Quotient<\/strong>: Dividing numbers translates to dividing their limits, just watch out for a zero denominator.<\/li>\n<li><strong>Powers and Roots<\/strong>: Raising a number to a power or taking a root? Apply the same operation to the limit.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572169042\">Show Solution<\/button><\/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\" style=\"text-align: center;\">[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<\/section>\n<section class=\"textbox example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572506406\">Show Solution<\/button><\/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<\/section>\n<div id=\"fs-id1170571675270\" class=\"textbook key-takeaways\">\n<h3>Limits of Polynomial and Rational Functions<\/h3>\n<p class=\"font-claude-response-body whitespace-normal break-words\">When working with polynomial and rational functions, you can often find limits by direct substitution.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Consider the limit [latex]\\underset{x\\to 2}{\\text{lim}}\\dfrac{2x^2-3x+1}{x^3+4}[\/latex]. Using limit laws step-by-step gives us [latex]\\frac{1}{4}[\/latex]. But notice what happens if we simply substitute [latex]x = 2[\/latex] directly into the function:<\/p>\n<p style=\"text-align: center;\">[latex]f(2) = \\dfrac{2(2)^2-3(2)+1}{(2)^3+4} = \\dfrac{8-6+1}{8+4} = \\dfrac{3}{12} = \\frac{1}{4}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">We get the same answer.<\/p>\n<p id=\"fs-id1170572133214\">This is not mere chance. It demonstrates a foundational concept in calculus: for polynomial and rational functions that are continuous at a point [latex]a[\/latex], the limit as [latex]x[\/latex] approaches [latex]a[\/latex] equals the value of the function at [latex]a[\/latex], or [latex]f(a)[\/latex]. This holds true provided the function is defined at that point.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>limits of polynomial and rational functions<\/h3>\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<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<\/section>\n<section class=\"textbox questionHelp\">\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].<\/p>\n<p>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 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<\/section>\n<section class=\"textbox example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572305892\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572305892\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572305892\">Since [latex]3[\/latex] 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 [latex]3[\/latex] for [latex]x[\/latex] into the function.<\/p>\n<p style=\"text-align: center;\">[latex]f(3)=\\frac{2(3)^2-3(3)+1}{5(3)+4} = \\dfrac{10}{19}[\/latex]<\/p>\n<p>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<\/section>\n<div>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm288276\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288276&theme=lumen&iframe_resize_id=ohm288276&source=tnh\" width=\"100%\" 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