{"id":1447,"date":"2023-06-05T14:51:50","date_gmt":"2023-06-05T14:51:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/finding-limits-properties-of-limits\/"},"modified":"2023-06-05T14:51:50","modified_gmt":"2023-06-05T14:51:50","slug":"finding-limits-properties-of-limits","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/finding-limits-properties-of-limits\/","title":{"raw":"Finding Limits: Properties of Limits","rendered":"Finding Limits: Properties of Limits"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li style=\"font-weight: 400;\">Find the limit of a sum, a difference, and a product.<\/li>\n \t<li style=\"font-weight: 400;\">Find the limit of a polynomial.<\/li>\n \t<li style=\"font-weight: 400;\">Find the limit of a power or a root.<\/li>\n \t<li style=\"font-weight: 400;\">Find the limit of a quotient.<\/li>\n<\/ul>\n<\/div>\n<h2>Finding the Limit of a Sum, a Difference, and a Product<\/h2>\nGraphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the <strong>properties of limits<\/strong>, which is a collection of theorems for finding limits.\n\nKnowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.\n<div class=\"textbox\">\n<h3>A General Note: Properties of Limits<\/h3>\nLet [latex]a,k,A[\/latex], and [latex]B[\/latex] represent real numbers, and [latex]f[\/latex] and [latex]g[\/latex] be functions, such that [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=A[\/latex] and [latex]\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=B[\/latex]. For limits that exist and are finite, the properties of limits are summarized in the table below.\n<table>\n<tbody>\n<tr>\n<td>Constant, <em>k<\/em><\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}k=k[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Constant times a function<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[k\\cdot f\\left(x\\right)\\right]=k\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=kA[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Sum of functions<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)+g\\left(x\\right)\\right]=\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)+\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=A+B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Difference of functions<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)-g\\left(x\\right)\\right]=\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)-\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=A-B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Product of functions<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)\\cdot g\\left(x\\right)\\right]=\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)\\cdot \\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=A\\cdot B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Quotient of functions<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\dfrac{f\\left(x\\right)}{g\\left(x\\right)}=\\dfrac{\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)}{\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)}=\\dfrac{A}{B},B\\ne 0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Function raised to an exponent<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}{\\left[f\\left(x\\right)\\right]}^{n}={\\left[\\underset{x\\to \\infty }{\\mathrm{lim}}f\\left(x\\right)\\right]}^{n}={A}^{n}[\/latex], where [latex]n[\/latex] is a positive integer<\/td>\n<\/tr>\n<tr>\n<td><em>n<\/em>th root of a function, where n is a positive integer<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\sqrt[n]{f\\left(x\\right)}=\\sqrt[n]{\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)\\right]}=\\sqrt[n]{A}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Polynomial function<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}p\\left(x\\right)=p\\left(a\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Evaluating the Limit of a Function Algebraically<\/h3>\nEvaluate [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(2x+5\\right)[\/latex].\n\n[reveal-answer q=\"159304\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"159304\"]\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 3}{\\mathrm{lim}}\\left(2x+5\\right)&amp;=\\underset{x\\to 3}{\\mathrm{lim}}\\left(2x\\right)+\\underset{x\\to 3}{\\mathrm{lim}}\\left(5\\right)&amp;&amp; \\text{Sum of functions property} \\\\ &amp;=\\underset{x\\to 3}{2\\mathrm{lim}}\\left(x\\right)+\\underset{x\\to 3}{\\mathrm{lim}}\\left(5\\right)&amp;&amp; \\text{Constant times a function property} \\\\ &amp;=2\\left(3\\right)+5 &amp;&amp; \\text{Evaluate} \\\\ &amp;=11 \\end{align}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nEvaluate the following limit: [latex]\\underset{x\\to -12}{\\mathrm{lim}}\\left(-2x+2\\right)[\/latex].\n\n[reveal-answer q=\"552188\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"552188\"]\n\n26\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]174089[\/ohm_question]\n\n<\/div>\n<h2>Finding the Limit of a Polynomial<\/h2>\nNot all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the <strong>limit<\/strong> of a polynomial function as [latex]x[\/latex] approaches [latex]a[\/latex] is equivalent to simply evaluating the function for [latex]a[\/latex] .\n<div class=\"textbox\">\n<h3>How To: Given a function containing a polynomial, find its limit.<\/h3>\n<ol>\n \t<li>Use the properties of limits to break up the polynomial into individual terms.<\/li>\n \t<li>Find the limits of the individual terms.<\/li>\n \t<li>Add the limits together.<\/li>\n \t<li>Alternatively, evaluate the function for [latex]a[\/latex] .<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Evaluating the Limit of a Function Algebraically<\/h3>\nEvaluate [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(5{x}^{2}\\right)[\/latex].\n\n[reveal-answer q=\"594542\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"594542\"]\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 3}{\\mathrm{lim}}\\left(5{x}^{2}\\right)&amp;=5\\underset{x\\to 3}{\\mathrm{lim}}\\left({x}^{2}\\right)&amp;&amp; \\text{Constant times a function property} \\\\ &amp;=5\\left({3}^{2}\\right)&amp;&amp; \\text{Function raised to an exponent property} \\\\ &amp;=45 \\end{align}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nEvaluate [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\left({x}^{3}-5\\right)[\/latex].\n\n[reveal-answer q=\"783831\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"783831\"]\n\n59\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Evaluating the Limit of a Polynomial Algebraically<\/h3>\nEvaluate [latex]\\underset{x\\to 5}{\\mathrm{lim}}\\left(2{x}^{3}-3x+1\\right)[\/latex].\n\n[reveal-answer q=\"100669\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"100669\"]\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 5}{\\mathrm{lim}}\\left(2{x}^{3}-3x+1\\right)&amp;=\\underset{x\\to 5}{\\mathrm{lim}}\\left(2{x}^{3}\\right)-\\underset{x\\to 5}{\\mathrm{lim}}\\left(3x\\right)+\\underset{x\\to 5}{\\mathrm{lim}}\\left(1\\right)&amp;&amp; \\text{Sum of functions} \\\\ &amp;=\\underset{x\\to 5}{2\\mathrm{lim}}\\left({x}^{3}\\right)-\\underset{x\\to 5}{3\\mathrm{lim}}\\left(x\\right)+\\underset{x\\to 5}{\\mathrm{lim}}\\left(1\\right)&amp;&amp; \\text{Constant times a function} \\\\ &amp;=2\\left({5}^{3}\\right)-3\\left(5\\right)+1&amp;&amp; \\text{Function raised to an exponent} \\\\ &amp;=236&amp;&amp; \\text{Evaluate} \\end{align}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nEvaluate the following limit: [latex]\\underset{x\\to -1}{\\mathrm{lim}}\\left({x}^{4}-4{x}^{3}+5\\right)[\/latex].\n\n[reveal-answer q=\"535807\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"535807\"]\n\n10\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]174090[\/ohm_question]\n\n<\/div>\n<h2>Finding the Limit of a Power or a Root<\/h2>\nWhen a limit includes a power or a root, we need another property to help us evaluate it. The square of the <strong>limit<\/strong> of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots.\n<div class=\"textbox shaded\">\n<h3>Example 4: Evaluating a Limit of a Power<\/h3>\nEvaluate [latex]\\underset{x\\to 2}{\\mathrm{lim}}{\\left(3x+1\\right)}^{5}[\/latex].\n\n[reveal-answer q=\"887017\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"887017\"]\n\nWe will take the limit of the function as [latex]x[\/latex] approaches 2 and raise the result to the 5<sup>th<\/sup> power.\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 2}{\\mathrm{lim}}{\\left(3x+1\\right)}^{5}&amp;={\\left(\\underset{x\\to 2}{\\mathrm{lim}}\\left(3x+1\\right)\\right)}^{5} \\\\ &amp;={\\left(3\\left(2\\right)+1\\right)}^{5} \\\\ &amp;={7}^{5} \\\\ &amp;=\\text{16,807} \\end{align}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nEvaluate the following limit: [latex]\\underset{x\\to -4}{\\mathrm{lim}}{\\left(10x+36\\right)}^{3}[\/latex].\n\n[reveal-answer q=\"411460\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"411460\"]\n\n-64\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>If we can\u2019t directly apply the properties of a limit, for example in [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\frac{{x}^{2}+6x+8}{x - 2}\\right)[\/latex] , can we still determine the limit of the function as [latex]x[\/latex] approaches [latex]a[\/latex] ?<\/h3>\n<em>Yes. Some functions may be algebraically rearranged so that one can evaluate the limit of a simplified equivalent form of the function.<\/em>\n\n<\/div>\n<h2>Finding the Limit of a Quotient<\/h2>\nFinding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.\n<div class=\"textbox\">\n<h3>How To: Given the limit of a function in quotient form, use factoring to evaluate it.<\/h3>\n<ol>\n \t<li>Factor the numerator and denominator completely.<\/li>\n \t<li>Simplify by dividing any factors common to the numerator and denominator.<\/li>\n \t<li>Evaluate the resulting limit, remembering to use the correct domain.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 5: Evaluating the Limit of a Quotient by Factoring<\/h3>\nEvaluate [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-6x+8}{x - 2}\\right)[\/latex].\n\n[reveal-answer q=\"592817\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"592817\"]\n\nFactor where possible, and simplify.\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\frac{{x}^{2}-6x+8}{x - 2}\\right)&amp;=\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\frac{\\left(x - 2\\right)\\left(x - 4\\right)}{x - 2}\\right)&amp;&amp; \\text{Factor the numerator}. \\\\ &amp;=\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\frac{\\cancel{\\left(x - 2\\right)}\\left(x - 4\\right)}{\\cancel{x - 2}}\\right)&amp;&amp; \\text{Cancel the common factors}. \\\\ &amp;=\\underset{x\\to 2}{\\mathrm{lim}}\\left(x - 4\\right)&amp;&amp; \\text{Evaluate}. \\\\ &amp;=2 - 4=-2 \\end{align}[\/latex]<\/p>\n\n<h4>Analysis of the Solution<\/h4>\nWhen the limit of a rational function cannot be evaluated directly, factored forms of the numerator and denominator may simplify to a result that can be evaluated.\n\nNotice, the function\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{{x}^{2}-6x+8}{x - 2}[\/latex]<\/p>\nis equivalent to the function\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=x - 4,x\\ne 2[\/latex].<\/p>\nNotice that the limit exists even though the function is not defined at [latex]x=2[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nEvaluate the following limit: [latex]\\underset{x\\to 7}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-11x+28}{7-x}\\right)[\/latex].\n\n[reveal-answer q=\"97030\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"97030\"]\n\n-3\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]16090[\/ohm_question]\n\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 6: Evaluating the Limit of a Quotient by Finding the LCD<\/h3>\nEvaluate [latex]\\underset{x\\to 5}{\\mathrm{lim}}\\left(\\dfrac{\\frac{1}{x}-\\frac{1}{5}}{x - 5}\\right)[\/latex].\n\n[reveal-answer q=\"649722\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"649722\"]\n\nFind the LCD for the denominators of the two terms in the numerator, and convert both fractions to have the LCD as their denominator.\n\n[caption id=\"\" align=\"aligncenter\" width=\"603\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185258\/CNX_Precalc_EQ_12_02_0012.jpg\" alt=\"Multiply numerator and denominator by LCD. Apply distributive property. Simplify. Factor the numerator. Cancel out like fractions. Evaluate for x=5.\" width=\"603\" height=\"450\"> <b>Figure 3<\/b>[\/caption]\n<h4>Analysis of the Solution<\/h4>\nWhen determining the <strong>limit<\/strong> of a rational function that has terms added or subtracted in either the numerator or denominator, the first step is to find the common denominator of the added or subtracted terms; then, convert both terms to have that denominator, or simplify the rational function by multiplying numerator and denominator by the least common denominator. Then check to see if the resulting numerator and denominator have any common factors.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nEvaluate [latex]\\underset{x\\to -5}{\\mathrm{lim}}\\left(\\dfrac{\\frac{1}{5}+\\frac{1}{x}}{10+2x}\\right)[\/latex].\n\n[reveal-answer q=\"934779\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"934779\"]\n\n[latex]-\\frac{1}{50}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a limit of a function containing a root, use a conjugate to evaluate.<\/h3>\n<ol>\n \t<li>If the quotient as given is not in indeterminate [latex]\\left(\\frac{0}{0}\\right)[\/latex] form, evaluate directly.<\/li>\n \t<li>Otherwise, rewrite the sum (or difference) of two quotients as a single quotient, using the <strong>least common denominator (LCD)<\/strong>.<\/li>\n \t<li>If the numerator includes a root, rationalize the numerator; multiply the numerator and denominator by the <strong>conjugate<\/strong> of the numerator. Recall that [latex]a\\pm \\sqrt{b}[\/latex] are conjugates.<\/li>\n \t<li>Simplify.<\/li>\n \t<li>Evaluate the resulting limit.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 7: Evaluating a Limit Containing a Root Using a Conjugate<\/h3>\nEvaluate [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{25-x}-5}{x}\\right)[\/latex].\n\n[reveal-answer q=\"167031\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"167031\"]\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{\\sqrt{25-x}-5}{x}\\right)&amp;=\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{\\left(\\sqrt{25-x}-5\\right)}{x}\\cdot \\frac{\\left(\\sqrt{25-x}+5\\right)}{\\left(\\sqrt{25-x}+5\\right)}\\right)&amp;&amp; \\text{Multiply numerator and denominator by the conjugate}. \\\\ &amp;=\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{\\left(25-x\\right)-25}{x\\left(\\sqrt{25-x}+5\\right)}\\right)&amp;&amp; \\text{Multiply: }\\left(\\sqrt{25-x}-5\\right)\\cdot \\left(\\sqrt{25-x}+5\\right)=\\left(25-x\\right)-25. \\\\ &amp;=\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{-x}{x\\left(\\sqrt{25-x}+5\\right)}\\right)&amp;&amp; \\text{Combine like terms}. \\\\ &amp;=\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{-\\cancel{x}}{\\cancel{x}\\left(\\sqrt{25-x}+5\\right)}\\right)&amp;&amp; \\text{Simplify }\\frac{-x}{x}=-1. \\\\ &amp;=\\frac{-1}{\\sqrt{25 - 0}+5}&amp;&amp; \\text{Evaluate}. \\\\ &amp;=\\frac{-1}{5+5}=-\\frac{1}{10} \\end{align}[\/latex]<\/p>\n\n<h4>Analysis of the Solution<\/h4>\nWhen determining a <strong>limit<\/strong> of a function with a root as one of two terms where we cannot evaluate directly, think about multiplying the numerator and denominator by the conjugate of the terms.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nEvaluate the following limit: [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{16-h}-4}{h}\\right)[\/latex].\n\n[reveal-answer q=\"112150\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"112150\"]\n\n[latex]-\\frac{1}{8}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]174096[\/ohm_question]\n\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 8: Evaluating the Limit of a Quotient of a Function by Factoring<\/h3>\nEvaluate [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\dfrac{4-x}{\\sqrt{x}-2}\\right)[\/latex].\n\n[reveal-answer q=\"376512\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"376512\"]\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\frac{4-x}{\\sqrt{x}-2}\\right)&amp;=\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\frac{\\left(2+\\sqrt{x}\\right)\\left(2-\\sqrt{x}\\right)}{\\sqrt{x}-2}\\right)&amp;&amp; \\text{Factor.} \\\\ &amp;=\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\frac{\\left(2+\\sqrt{x}\\right)\\cancel{\\left(2-\\sqrt{x}\\right)}}{-\\cancel{\\left(2-\\sqrt{x}\\right)}}\\right)&amp;&amp; \\text{Factor }-1\\text{ out of the denominator and implify}. \\\\ &amp;=\\underset{x\\to 4}{\\mathrm{lim}}-\\left(2+\\sqrt{x}\\right)&amp;&amp; \\text{Evaluate}. \\\\ &amp;=-\\left(2+\\sqrt{4}\\right) \\\\ &amp;=-4 \\end{align}[\/latex]<\/p>\n\n<h4>Analysis of the Solution<\/h4>\nMultiplying by a conjugate would expand the numerator; look instead for factors in the numerator. Four is a perfect square so that the numerator is in the form\n<p style=\"text-align: center;\">[latex]{a}^{2}-{b}^{2}[\/latex]<\/p>\nand may be factored as\n<p style=\"text-align: center;\">[latex]\\left(a+b\\right)\\left(a-b\\right)[\/latex].<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nEvaluate the following limit: [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(\\dfrac{x - 3}{\\sqrt{x}-\\sqrt{3}}\\right)[\/latex].\n\n[reveal-answer q=\"591987\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"591987\"]\n\n[latex]2\\sqrt{3}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a quotient with absolute values, evaluate its limit.<\/h3>\n<ol>\n \t<li>Try factoring or finding the LCD.<\/li>\n \t<li>If the <strong>limit<\/strong> cannot be found, choose several values close to and on either side of the input where the function is undefined.<\/li>\n \t<li>Use the numeric evidence to estimate the limits on both sides.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 9: Evaluating the Limit of a Quotient with Absolute Values<\/h3>\nEvaluate [latex]\\underset{x\\to 7}{\\mathrm{lim}}\\dfrac{|x - 7|}{x - 7}[\/latex].\n\n[reveal-answer q=\"889215\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"889215\"]\n\nThe function is undefined at [latex]x=7[\/latex], so we will try values close to 7 from the left and the right.\n\nLeft-hand limit: [latex]\\frac{|6.9 - 7|}{6.9 - 7}=\\frac{|6.99 - 7|}{6.99 - 7}=\\frac{|6.999 - 7|}{6.999 - 7}=-1[\/latex]\n\nRight-hand limit: [latex]\\frac{|7.1 - 7|}{7.1 - 7}=\\frac{|7.01 - 7|}{7.01 - 7}=\\frac{|7.001 - 7|}{7.001 - 7}=1[\/latex]\n\nSince the left- and right-hand limits are not equal, there is no limit.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nEvaluate [latex]\\underset{x\\to {6}^{+}}{\\mathrm{lim}}\\dfrac{6-x}{|x - 6|}[\/latex].\n\n[reveal-answer q=\"496056\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"496056\"]\n\n-1\n\n[\/hidden-answer]\n\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>The properties of limits can be used to perform operations on the limits of functions rather than the functions themselves.<\/li>\n \t<li>The limit of a polynomial function can be found by finding the sum of the limits of the individual terms.<\/li>\n \t<li>The limit of a function that has been raised to a power equals the same power of the limit of the function. Another method is direct substitution.<\/li>\n \t<li>The limit of the root of a function equals the corresponding root of the limit of the function.<\/li>\n \t<li>One way to find the limit of a function expressed as a quotient is to write the quotient in factored form and simplify.<\/li>\n \t<li>Another method of finding the limit of a complex fraction is to find the LCD.<\/li>\n \t<li>A limit containing a function containing a root may be evaluated using a conjugate.<\/li>\n \t<li>The limits of some functions expressed as quotients can be found by factoring.<\/li>\n \t<li>One way to evaluate the limit of a quotient containing absolute values is by using numeric evidence. Setting it up piecewise can also be useful.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135453295\" class=\"definition\">\n \t<dt>properties of limits<\/dt>\n \t<dd id=\"fs-id1165135453300\">a collection of theorems for finding limits of functions by performing mathematical operations on the limits<\/dd>\n<\/dl>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400;\">Find the limit of a sum, a difference, and a product.<\/li>\n<li style=\"font-weight: 400;\">Find the limit of a polynomial.<\/li>\n<li style=\"font-weight: 400;\">Find the limit of a power or a root.<\/li>\n<li style=\"font-weight: 400;\">Find the limit of a quotient.<\/li>\n<\/ul>\n<\/div>\n<h2>Finding the Limit of a Sum, a Difference, and a Product<\/h2>\n<p>Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the <strong>properties of limits<\/strong>, which is a collection of theorems for finding limits.<\/p>\n<p>Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Properties of Limits<\/h3>\n<p>Let [latex]a,k,A[\/latex], and [latex]B[\/latex] represent real numbers, and [latex]f[\/latex] and [latex]g[\/latex] be functions, such that [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=A[\/latex] and [latex]\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=B[\/latex]. For limits that exist and are finite, the properties of limits are summarized in the table below.<\/p>\n<table>\n<tbody>\n<tr>\n<td>Constant, <em>k<\/em><\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}k=k[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Constant times a function<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[k\\cdot f\\left(x\\right)\\right]=k\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=kA[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Sum of functions<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)+g\\left(x\\right)\\right]=\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)+\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=A+B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Difference of functions<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)-g\\left(x\\right)\\right]=\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)-\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=A-B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Product of functions<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)\\cdot g\\left(x\\right)\\right]=\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)\\cdot \\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)=A\\cdot B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Quotient of functions<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\dfrac{f\\left(x\\right)}{g\\left(x\\right)}=\\dfrac{\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)}{\\underset{x\\to a}{\\mathrm{lim}}g\\left(x\\right)}=\\dfrac{A}{B},B\\ne 0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Function raised to an exponent<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}{\\left[f\\left(x\\right)\\right]}^{n}={\\left[\\underset{x\\to \\infty }{\\mathrm{lim}}f\\left(x\\right)\\right]}^{n}={A}^{n}[\/latex], where [latex]n[\/latex] is a positive integer<\/td>\n<\/tr>\n<tr>\n<td><em>n<\/em>th root of a function, where n is a positive integer<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}\\sqrt[n]{f\\left(x\\right)}=\\sqrt[n]{\\underset{x\\to a}{\\mathrm{lim}}\\left[f\\left(x\\right)\\right]}=\\sqrt[n]{A}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Polynomial function<\/td>\n<td>[latex]\\underset{x\\to a}{\\mathrm{lim}}p\\left(x\\right)=p\\left(a\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Evaluating the Limit of a Function Algebraically<\/h3>\n<p>Evaluate [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(2x+5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q159304\">Show Solution<\/span><\/p>\n<div id=\"q159304\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 3}{\\mathrm{lim}}\\left(2x+5\\right)&=\\underset{x\\to 3}{\\mathrm{lim}}\\left(2x\\right)+\\underset{x\\to 3}{\\mathrm{lim}}\\left(5\\right)&& \\text{Sum of functions property} \\\\ &=\\underset{x\\to 3}{2\\mathrm{lim}}\\left(x\\right)+\\underset{x\\to 3}{\\mathrm{lim}}\\left(5\\right)&& \\text{Constant times a function property} \\\\ &=2\\left(3\\right)+5 && \\text{Evaluate} \\\\ &=11 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Evaluate the following limit: [latex]\\underset{x\\to -12}{\\mathrm{lim}}\\left(-2x+2\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q552188\">Show Solution<\/span><\/p>\n<div id=\"q552188\" class=\"hidden-answer\" style=\"display: none\">\n<p>26<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174089\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174089&theme=oea&iframe_resize_id=ohm174089\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Finding the Limit of a Polynomial<\/h2>\n<p>Not all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the <strong>limit<\/strong> of a polynomial function as [latex]x[\/latex] approaches [latex]a[\/latex] is equivalent to simply evaluating the function for [latex]a[\/latex] .<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a function containing a polynomial, find its limit.<\/h3>\n<ol>\n<li>Use the properties of limits to break up the polynomial into individual terms.<\/li>\n<li>Find the limits of the individual terms.<\/li>\n<li>Add the limits together.<\/li>\n<li>Alternatively, evaluate the function for [latex]a[\/latex] .<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Evaluating the Limit of a Function Algebraically<\/h3>\n<p>Evaluate [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(5{x}^{2}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q594542\">Show Solution<\/span><\/p>\n<div id=\"q594542\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 3}{\\mathrm{lim}}\\left(5{x}^{2}\\right)&=5\\underset{x\\to 3}{\\mathrm{lim}}\\left({x}^{2}\\right)&& \\text{Constant times a function property} \\\\ &=5\\left({3}^{2}\\right)&& \\text{Function raised to an exponent property} \\\\ &=45 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Evaluate [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\left({x}^{3}-5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783831\">Show Solution<\/span><\/p>\n<div id=\"q783831\" class=\"hidden-answer\" style=\"display: none\">\n<p>59<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Evaluating the Limit of a Polynomial Algebraically<\/h3>\n<p>Evaluate [latex]\\underset{x\\to 5}{\\mathrm{lim}}\\left(2{x}^{3}-3x+1\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q100669\">Show Solution<\/span><\/p>\n<div id=\"q100669\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 5}{\\mathrm{lim}}\\left(2{x}^{3}-3x+1\\right)&=\\underset{x\\to 5}{\\mathrm{lim}}\\left(2{x}^{3}\\right)-\\underset{x\\to 5}{\\mathrm{lim}}\\left(3x\\right)+\\underset{x\\to 5}{\\mathrm{lim}}\\left(1\\right)&& \\text{Sum of functions} \\\\ &=\\underset{x\\to 5}{2\\mathrm{lim}}\\left({x}^{3}\\right)-\\underset{x\\to 5}{3\\mathrm{lim}}\\left(x\\right)+\\underset{x\\to 5}{\\mathrm{lim}}\\left(1\\right)&& \\text{Constant times a function} \\\\ &=2\\left({5}^{3}\\right)-3\\left(5\\right)+1&& \\text{Function raised to an exponent} \\\\ &=236&& \\text{Evaluate} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Evaluate the following limit: [latex]\\underset{x\\to -1}{\\mathrm{lim}}\\left({x}^{4}-4{x}^{3}+5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q535807\">Show Solution<\/span><\/p>\n<div id=\"q535807\" class=\"hidden-answer\" style=\"display: none\">\n<p>10<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174090\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174090&theme=oea&iframe_resize_id=ohm174090\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Finding the Limit of a Power or a Root<\/h2>\n<p>When a limit includes a power or a root, we need another property to help us evaluate it. The square of the <strong>limit<\/strong> of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots.<\/p>\n<div class=\"textbox shaded\">\n<h3>Example 4: Evaluating a Limit of a Power<\/h3>\n<p>Evaluate [latex]\\underset{x\\to 2}{\\mathrm{lim}}{\\left(3x+1\\right)}^{5}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q887017\">Show Solution<\/span><\/p>\n<div id=\"q887017\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will take the limit of the function as [latex]x[\/latex] approaches 2 and raise the result to the 5<sup>th<\/sup> power.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 2}{\\mathrm{lim}}{\\left(3x+1\\right)}^{5}&={\\left(\\underset{x\\to 2}{\\mathrm{lim}}\\left(3x+1\\right)\\right)}^{5} \\\\ &={\\left(3\\left(2\\right)+1\\right)}^{5} \\\\ &={7}^{5} \\\\ &=\\text{16,807} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Evaluate the following limit: [latex]\\underset{x\\to -4}{\\mathrm{lim}}{\\left(10x+36\\right)}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q411460\">Show Solution<\/span><\/p>\n<div id=\"q411460\" class=\"hidden-answer\" style=\"display: none\">\n<p>-64<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>If we can\u2019t directly apply the properties of a limit, for example in [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\frac{{x}^{2}+6x+8}{x - 2}\\right)[\/latex] , can we still determine the limit of the function as [latex]x[\/latex] approaches [latex]a[\/latex] ?<\/h3>\n<p><em>Yes. Some functions may be algebraically rearranged so that one can evaluate the limit of a simplified equivalent form of the function.<\/em><\/p>\n<\/div>\n<h2>Finding the Limit of a Quotient<\/h2>\n<p>Finding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the limit of a function in quotient form, use factoring to evaluate it.<\/h3>\n<ol>\n<li>Factor the numerator and denominator completely.<\/li>\n<li>Simplify by dividing any factors common to the numerator and denominator.<\/li>\n<li>Evaluate the resulting limit, remembering to use the correct domain.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 5: Evaluating the Limit of a Quotient by Factoring<\/h3>\n<p>Evaluate [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-6x+8}{x - 2}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q592817\">Show Solution<\/span><\/p>\n<div id=\"q592817\" class=\"hidden-answer\" style=\"display: none\">\n<p>Factor where possible, and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\frac{{x}^{2}-6x+8}{x - 2}\\right)&=\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\frac{\\left(x - 2\\right)\\left(x - 4\\right)}{x - 2}\\right)&& \\text{Factor the numerator}. \\\\ &=\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\frac{\\cancel{\\left(x - 2\\right)}\\left(x - 4\\right)}{\\cancel{x - 2}}\\right)&& \\text{Cancel the common factors}. \\\\ &=\\underset{x\\to 2}{\\mathrm{lim}}\\left(x - 4\\right)&& \\text{Evaluate}. \\\\ &=2 - 4=-2 \\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>When the limit of a rational function cannot be evaluated directly, factored forms of the numerator and denominator may simplify to a result that can be evaluated.<\/p>\n<p>Notice, the function<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\frac{{x}^{2}-6x+8}{x - 2}[\/latex]<\/p>\n<p>is equivalent to the function<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=x - 4,x\\ne 2[\/latex].<\/p>\n<p>Notice that the limit exists even though the function is not defined at [latex]x=2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Evaluate the following limit: [latex]\\underset{x\\to 7}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-11x+28}{7-x}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q97030\">Show Solution<\/span><\/p>\n<div id=\"q97030\" class=\"hidden-answer\" style=\"display: none\">\n<p>-3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm16090\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=16090&theme=oea&iframe_resize_id=ohm16090\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 6: Evaluating the Limit of a Quotient by Finding the LCD<\/h3>\n<p>Evaluate [latex]\\underset{x\\to 5}{\\mathrm{lim}}\\left(\\dfrac{\\frac{1}{x}-\\frac{1}{5}}{x - 5}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q649722\">Show Solution<\/span><\/p>\n<div id=\"q649722\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the LCD for the denominators of the two terms in the numerator, and convert both fractions to have the LCD as their denominator.<\/p>\n<div style=\"width: 613px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185258\/CNX_Precalc_EQ_12_02_0012.jpg\" alt=\"Multiply numerator and denominator by LCD. Apply distributive property. Simplify. Factor the numerator. Cancel out like fractions. Evaluate for x=5.\" width=\"603\" height=\"450\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>When determining the <strong>limit<\/strong> of a rational function that has terms added or subtracted in either the numerator or denominator, the first step is to find the common denominator of the added or subtracted terms; then, convert both terms to have that denominator, or simplify the rational function by multiplying numerator and denominator by the least common denominator. Then check to see if the resulting numerator and denominator have any common factors.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Evaluate [latex]\\underset{x\\to -5}{\\mathrm{lim}}\\left(\\dfrac{\\frac{1}{5}+\\frac{1}{x}}{10+2x}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q934779\">Show Solution<\/span><\/p>\n<div id=\"q934779\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-\\frac{1}{50}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a limit of a function containing a root, use a conjugate to evaluate.<\/h3>\n<ol>\n<li>If the quotient as given is not in indeterminate [latex]\\left(\\frac{0}{0}\\right)[\/latex] form, evaluate directly.<\/li>\n<li>Otherwise, rewrite the sum (or difference) of two quotients as a single quotient, using the <strong>least common denominator (LCD)<\/strong>.<\/li>\n<li>If the numerator includes a root, rationalize the numerator; multiply the numerator and denominator by the <strong>conjugate<\/strong> of the numerator. Recall that [latex]a\\pm \\sqrt{b}[\/latex] are conjugates.<\/li>\n<li>Simplify.<\/li>\n<li>Evaluate the resulting limit.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 7: Evaluating a Limit Containing a Root Using a Conjugate<\/h3>\n<p>Evaluate [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{25-x}-5}{x}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q167031\">Show Solution<\/span><\/p>\n<div id=\"q167031\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{\\sqrt{25-x}-5}{x}\\right)&=\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{\\left(\\sqrt{25-x}-5\\right)}{x}\\cdot \\frac{\\left(\\sqrt{25-x}+5\\right)}{\\left(\\sqrt{25-x}+5\\right)}\\right)&& \\text{Multiply numerator and denominator by the conjugate}. \\\\ &=\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{\\left(25-x\\right)-25}{x\\left(\\sqrt{25-x}+5\\right)}\\right)&& \\text{Multiply: }\\left(\\sqrt{25-x}-5\\right)\\cdot \\left(\\sqrt{25-x}+5\\right)=\\left(25-x\\right)-25. \\\\ &=\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{-x}{x\\left(\\sqrt{25-x}+5\\right)}\\right)&& \\text{Combine like terms}. \\\\ &=\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{-\\cancel{x}}{\\cancel{x}\\left(\\sqrt{25-x}+5\\right)}\\right)&& \\text{Simplify }\\frac{-x}{x}=-1. \\\\ &=\\frac{-1}{\\sqrt{25 - 0}+5}&& \\text{Evaluate}. \\\\ &=\\frac{-1}{5+5}=-\\frac{1}{10} \\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>When determining a <strong>limit<\/strong> of a function with a root as one of two terms where we cannot evaluate directly, think about multiplying the numerator and denominator by the conjugate of the terms.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Evaluate the following limit: [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{16-h}-4}{h}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q112150\">Show Solution<\/span><\/p>\n<div id=\"q112150\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-\\frac{1}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174096\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174096&theme=oea&iframe_resize_id=ohm174096\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 8: Evaluating the Limit of a Quotient of a Function by Factoring<\/h3>\n<p>Evaluate [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\dfrac{4-x}{\\sqrt{x}-2}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q376512\">Show Solution<\/span><\/p>\n<div id=\"q376512\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\frac{4-x}{\\sqrt{x}-2}\\right)&=\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\frac{\\left(2+\\sqrt{x}\\right)\\left(2-\\sqrt{x}\\right)}{\\sqrt{x}-2}\\right)&& \\text{Factor.} \\\\ &=\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\frac{\\left(2+\\sqrt{x}\\right)\\cancel{\\left(2-\\sqrt{x}\\right)}}{-\\cancel{\\left(2-\\sqrt{x}\\right)}}\\right)&& \\text{Factor }-1\\text{ out of the denominator and implify}. \\\\ &=\\underset{x\\to 4}{\\mathrm{lim}}-\\left(2+\\sqrt{x}\\right)&& \\text{Evaluate}. \\\\ &=-\\left(2+\\sqrt{4}\\right) \\\\ &=-4 \\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Multiplying by a conjugate would expand the numerator; look instead for factors in the numerator. Four is a perfect square so that the numerator is in the form<\/p>\n<p style=\"text-align: center;\">[latex]{a}^{2}-{b}^{2}[\/latex]<\/p>\n<p>and may be factored as<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+b\\right)\\left(a-b\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Evaluate the following limit: [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(\\dfrac{x - 3}{\\sqrt{x}-\\sqrt{3}}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q591987\">Show Solution<\/span><\/p>\n<div id=\"q591987\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]2\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a quotient with absolute values, evaluate its limit.<\/h3>\n<ol>\n<li>Try factoring or finding the LCD.<\/li>\n<li>If the <strong>limit<\/strong> cannot be found, choose several values close to and on either side of the input where the function is undefined.<\/li>\n<li>Use the numeric evidence to estimate the limits on both sides.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 9: Evaluating the Limit of a Quotient with Absolute Values<\/h3>\n<p>Evaluate [latex]\\underset{x\\to 7}{\\mathrm{lim}}\\dfrac{|x - 7|}{x - 7}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q889215\">Show Solution<\/span><\/p>\n<div id=\"q889215\" class=\"hidden-answer\" style=\"display: none\">\n<p>The function is undefined at [latex]x=7[\/latex], so we will try values close to 7 from the left and the right.<\/p>\n<p>Left-hand limit: [latex]\\frac{|6.9 - 7|}{6.9 - 7}=\\frac{|6.99 - 7|}{6.99 - 7}=\\frac{|6.999 - 7|}{6.999 - 7}=-1[\/latex]<\/p>\n<p>Right-hand limit: [latex]\\frac{|7.1 - 7|}{7.1 - 7}=\\frac{|7.01 - 7|}{7.01 - 7}=\\frac{|7.001 - 7|}{7.001 - 7}=1[\/latex]<\/p>\n<p>Since the left- and right-hand limits are not equal, there is no limit.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Evaluate [latex]\\underset{x\\to {6}^{+}}{\\mathrm{lim}}\\dfrac{6-x}{|x - 6|}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q496056\">Show Solution<\/span><\/p>\n<div id=\"q496056\" class=\"hidden-answer\" style=\"display: none\">\n<p>-1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>The properties of limits can be used to perform operations on the limits of functions rather than the functions themselves.<\/li>\n<li>The limit of a polynomial function can be found by finding the sum of the limits of the individual terms.<\/li>\n<li>The limit of a function that has been raised to a power equals the same power of the limit of the function. Another method is direct substitution.<\/li>\n<li>The limit of the root of a function equals the corresponding root of the limit of the function.<\/li>\n<li>One way to find the limit of a function expressed as a quotient is to write the quotient in factored form and simplify.<\/li>\n<li>Another method of finding the limit of a complex fraction is to find the LCD.<\/li>\n<li>A limit containing a function containing a root may be evaluated using a conjugate.<\/li>\n<li>The limits of some functions expressed as quotients can be found by factoring.<\/li>\n<li>One way to evaluate the limit of a quotient containing absolute values is by using numeric evidence. Setting it up piecewise can also be useful.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135453295\" class=\"definition\">\n<dt>properties of limits<\/dt>\n<dd id=\"fs-id1165135453300\">a collection of theorems for finding limits of functions by performing mathematical operations on the limits<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1447\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":503070,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1447","chapter","type-chapter","status-publish","hentry"],"part":1445,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1447","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/users\/503070"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1447\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/parts\/1445"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1447\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/media?parent=1447"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapter-type?post=1447"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/contributor?post=1447"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/license?post=1447"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}