{"id":3895,"date":"2021-05-20T18:29:56","date_gmt":"2021-05-20T18:29:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-limits-at-infinity-and-asymptotes\/"},"modified":"2021-07-03T19:05:56","modified_gmt":"2021-07-03T19:05:56","slug":"review-for-limits-at-infinity-and-asymptotes","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-limits-at-infinity-and-asymptotes\/","title":{"raw":"Skills Review for Limits at Infinity and Asymptotes","rendered":"Skills Review for Limits at Infinity and Asymptotes"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Divide polynomials by binomials using synthetic division or long division<\/li>\r\n \t<li>Identify horizontal asymptotes<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Limits at Infinity and Asymptotes section, skills needed include polynomial long division and finding horizontal asymptotes. These two topics will be reviewed here.\r\n<h2>Perform Polynomial Long Division<\/h2>\r\nDivision of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm, it would look like this:\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204321\/CNX_Precalc_revised_eq_12.png\" alt=\"Set up the division problem. 2x cubed divided by x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Then bring down the next term. Negative 7x squared divided by x is negative 7x. Multiply the sum of x and 2 by negative 7x. Subtract, then bring down the next term. 18x divided by x is 18. Multiply the sum of x and 2 by 18. Subtract.\" width=\"522\" height=\"462\" \/>\r\n\r\nWe have found\r\n<p style=\"text-align: center;\">[latex]\\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\\frac{31}{x+2}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">or<\/p>\r\n<p style=\"text-align: center;\">[latex]2{x}^{3}-3{x}^{2}+4x+5=\\left(x+2\\right)\\left(2{x}^{2}-7x+18\\right)-31[\/latex]<\/p>\r\n\r\n<div class=\"textbox\">\r\n<h3>How To: Given a polynomial and a binomial, use long division to divide the polynomial by the binomial<\/h3>\r\n<ol>\r\n \t<li>Set up the division problem.<\/li>\r\n \t<li>Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.<\/li>\r\n \t<li>Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\r\n \t<li>Subtract the bottom <strong>binomial<\/strong> from the terms above it.<\/li>\r\n \t<li>Bring down the next term of the dividend.<\/li>\r\n \t<li>Repeat steps 2\u20135 until reaching the last term of the dividend.<\/li>\r\n \t<li>If the remainder is non-zero, express as a fraction using the divisor as the denominator.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Long Division to Divide a Second-Degree Polynomial<\/h3>\r\nDivide [latex]5{x}^{2}+3x - 2[\/latex]\u00a0by [latex]x+1[\/latex].\r\n\r\n[reveal-answer q=\"996959\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"996959\"]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204327\/CNX_Precalc_revised_eq_22.png\" alt=\"Set up the division problem. 5x squared divided by x is 5x. Multiply x plus 1 by 5x. Subtract. Bring down the next term. Negative 2x divded by x is negative 2. Multiply x + 1 by negative 2. Subtract.\" width=\"426\" height=\"288\" \/>The quotient is [latex]5x - 2[\/latex].\u00a0The remainder is 0. We write the result as\r\n<p style=\"text-align: center;\">[latex]\\frac{5{x}^{2}+3x - 2}{x+1}=5x - 2[\/latex]<\/p>\r\nor\r\n<p style=\"text-align: center;\">[latex]5{x}^{2}+3x - 2=\\left(x+1\\right)\\left(5x - 2\\right)[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nThis division problem had a remainder of 0. This tells us that the dividend is divided evenly by the divisor and that the divisor is a factor of the dividend.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Long Division to Divide a Third-Degree Polynomial<\/h3>\r\nDivide [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]\u00a0by [latex]3x - 2[\/latex].\r\n\r\n[reveal-answer q=\"850001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"850001\"]\r\n\r\n<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/replacesquareroot.png\"><img class=\"aligncenter wp-image-11885\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204330\/replacesquareroot.png\" alt=\"6x cubed divided by 3x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Bring down the next term. 15x squared divided by 3x is 5x. Multiply 3x minus 2 by 5x. Subtract. Bring down the next term. Negative 21x divided by 3x is negative 7. Multiply 3x minus 2 by negative 7. Subtract. The remainder is 1.\" width=\"621\" height=\"153\" \/><\/a>\r\n\r\nThere is a remainder of 1. We can express the result as:\r\n\r\n[latex]\\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x - 2}=2{x}^{2}+5x - 7+\\frac{1}{3x - 2}[\/latex]\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can check our work by using the Division Algorithm to rewrite the solution then multiplying.\r\n\r\n[latex]\\left(3x - 2\\right)\\left(2{x}^{2}+5x - 7\\right)+1=6{x}^{3}+11{x}^{2}-31x+15[\/latex]\r\n\r\nNotice, as we write our result,\r\n<ul id=\"fs-id1165135152079\">\r\n \t<li>the dividend is [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/li>\r\n \t<li>the divisor is [latex]3x - 2[\/latex]<\/li>\r\n \t<li>the quotient is [latex]2{x}^{2}+5x - 7[\/latex]<\/li>\r\n \t<li>the remainder is\u00a01<\/li>\r\n<\/ul>\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]29482[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Find Horizontal Asymptotes<\/h2>\r\nHorizontal asymptotes help describe the behavior of a graph as the <em>input<\/em> gets very large or very small. Recall that a polynomial\u2019s end behavior will mirror that of the leading term. Likewise, a rational function\u2019s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Horizontal Asymptotes of Rational Functions<\/h3>\r\nThe <strong>horizontal asymptote<\/strong> of a rational function can be determined by looking at the degrees of the numerator and denominator.\r\n<ul id=\"fs-id1165137722720\">\r\n \t<li><strong>Case 1:<\/strong> Degree of numerator <em>is less than<\/em> degree of denominator: horizontal asymptote at\u00a0[latex]y=0[\/latex]<\/li>\r\n \t<li><strong>Case 2<\/strong>: Degree of numerator <em>is greater than degree of denominator by one<\/em>: no horizontal asymptote; slant asymptote.\r\n<ul>\r\n \t<li>If the degree of the numerator is greater than the degree of the denominator by\u00a0<em>more than one<\/em>, the end behavior of the function's graph will mimic that of the graph of the reduced ratio of leading terms.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Case 3<\/strong>: Degree of numerator <em>is equal to<\/em> degree of denominator: horizontal asymptote at ratio of leading coefficients.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying Horizontal and Slant Asymptotes<\/h3>\r\nFor the functions below, identify the horizontal or slant asymptote.\r\n<ol>\r\n \t<li>[latex]g\\left(x\\right)=\\dfrac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[\/latex]<\/li>\r\n \t<li>[latex]h\\left(x\\right)=\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex]<\/li>\r\n \t<li>[latex]k\\left(x\\right)=\\dfrac{{x}^{2}+4x}{{x}^{3}-8}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"968793\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"968793\"]\r\n\r\nFor these solutions, we will use [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)}, q\\left(x\\right)\\ne 0[\/latex].\r\n<ol>\r\n \t<li>[latex]g\\left(x\\right)=\\dfrac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[\/latex]: The degree of [latex]p[\/latex] and the degree of [latex]q[\/latex] are both equal to 3, so we can find the horizontal asymptote by taking the ratio of the leading terms. There is a horizontal asymptote at [latex]y=\\frac{6}{2}[\/latex] or [latex]y=3[\/latex].<\/li>\r\n \t<li>[latex]h\\left(x\\right)=\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex]: The degree of [latex]p=2[\/latex] and degree of [latex]q=1[\/latex]. Since [latex]p&gt;q[\/latex] by 1, there is a slant asymptote. Perform polynomial long divsion to find the slant asympote. The slant asymptote is [latex]y=-x - 6[\/latex].<\/li>\r\n \t<li>[latex]k\\left(x\\right)=\\dfrac{{x}^{2}+4x}{{x}^{3}-8}[\/latex]: The degree of [latex]p=2\\text{ }&lt;[\/latex] degree of [latex]q=3[\/latex], so there is a horizontal asymptote [latex]y=0[\/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]221906[\/ohm_question]\r\n\r\n<\/div>\r\nWatch this video to see more worked examples of determining which kind of horizontal asymptote a rational function will have.\r\n\r\nhttps:\/\/youtu.be\/A1tApZSE8nI","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Divide polynomials by binomials using synthetic division or long division<\/li>\n<li>Identify horizontal asymptotes<\/li>\n<\/ul>\n<\/div>\n<p>In the Limits at Infinity and Asymptotes section, skills needed include polynomial long division and finding horizontal asymptotes. These two topics will be reviewed here.<\/p>\n<h2>Perform Polynomial Long Division<\/h2>\n<p>Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm, it would look like this:<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204321\/CNX_Precalc_revised_eq_12.png\" alt=\"Set up the division problem. 2x cubed divided by x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Then bring down the next term. Negative 7x squared divided by x is negative 7x. Multiply the sum of x and 2 by negative 7x. Subtract, then bring down the next term. 18x divided by x is 18. Multiply the sum of x and 2 by 18. Subtract.\" width=\"522\" height=\"462\" \/><\/p>\n<p>We have found<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\\frac{31}{x+2}[\/latex]<\/p>\n<p style=\"text-align: center;\">or<\/p>\n<p style=\"text-align: center;\">[latex]2{x}^{3}-3{x}^{2}+4x+5=\\left(x+2\\right)\\left(2{x}^{2}-7x+18\\right)-31[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a polynomial and a binomial, use long division to divide the polynomial by the binomial<\/h3>\n<ol>\n<li>Set up the division problem.<\/li>\n<li>Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.<\/li>\n<li>Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\n<li>Subtract the bottom <strong>binomial<\/strong> from the terms above it.<\/li>\n<li>Bring down the next term of the dividend.<\/li>\n<li>Repeat steps 2\u20135 until reaching the last term of the dividend.<\/li>\n<li>If the remainder is non-zero, express as a fraction using the divisor as the denominator.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Long Division to Divide a Second-Degree Polynomial<\/h3>\n<p>Divide [latex]5{x}^{2}+3x - 2[\/latex]\u00a0by [latex]x+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q996959\">Show Solution<\/span><\/p>\n<div id=\"q996959\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204327\/CNX_Precalc_revised_eq_22.png\" alt=\"Set up the division problem. 5x squared divided by x is 5x. Multiply x plus 1 by 5x. Subtract. Bring down the next term. Negative 2x divded by x is negative 2. Multiply x + 1 by negative 2. Subtract.\" width=\"426\" height=\"288\" \/>The quotient is [latex]5x - 2[\/latex].\u00a0The remainder is 0. We write the result as<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5{x}^{2}+3x - 2}{x+1}=5x - 2[\/latex]<\/p>\n<p>or<\/p>\n<p style=\"text-align: center;\">[latex]5{x}^{2}+3x - 2=\\left(x+1\\right)\\left(5x - 2\\right)[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>This division problem had a remainder of 0. This tells us that the dividend is divided evenly by the divisor and that the divisor is a factor of the dividend.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Long Division to Divide a Third-Degree Polynomial<\/h3>\n<p>Divide [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]\u00a0by [latex]3x - 2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q850001\">Show Solution<\/span><\/p>\n<div id=\"q850001\" class=\"hidden-answer\" style=\"display: none\">\n<p><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/replacesquareroot.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-11885\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02204330\/replacesquareroot.png\" alt=\"6x cubed divided by 3x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Bring down the next term. 15x squared divided by 3x is 5x. Multiply 3x minus 2 by 5x. Subtract. Bring down the next term. Negative 21x divided by 3x is negative 7. Multiply 3x minus 2 by negative 7. Subtract. The remainder is 1.\" width=\"621\" height=\"153\" \/><\/a><\/p>\n<p>There is a remainder of 1. We can express the result as:<\/p>\n<p>[latex]\\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x - 2}=2{x}^{2}+5x - 7+\\frac{1}{3x - 2}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can check our work by using the Division Algorithm to rewrite the solution then multiplying.<\/p>\n<p>[latex]\\left(3x - 2\\right)\\left(2{x}^{2}+5x - 7\\right)+1=6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/p>\n<p>Notice, as we write our result,<\/p>\n<ul id=\"fs-id1165135152079\">\n<li>the dividend is [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/li>\n<li>the divisor is [latex]3x - 2[\/latex]<\/li>\n<li>the quotient is [latex]2{x}^{2}+5x - 7[\/latex]<\/li>\n<li>the remainder is\u00a01<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm29482\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29482&theme=oea&iframe_resize_id=ohm29482&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Find Horizontal Asymptotes<\/h2>\n<p>Horizontal asymptotes help describe the behavior of a graph as the <em>input<\/em> gets very large or very small. Recall that a polynomial\u2019s end behavior will mirror that of the leading term. Likewise, a rational function\u2019s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Horizontal Asymptotes of Rational Functions<\/h3>\n<p>The <strong>horizontal asymptote<\/strong> of a rational function can be determined by looking at the degrees of the numerator and denominator.<\/p>\n<ul id=\"fs-id1165137722720\">\n<li><strong>Case 1:<\/strong> Degree of numerator <em>is less than<\/em> degree of denominator: horizontal asymptote at\u00a0[latex]y=0[\/latex]<\/li>\n<li><strong>Case 2<\/strong>: Degree of numerator <em>is greater than degree of denominator by one<\/em>: no horizontal asymptote; slant asymptote.\n<ul>\n<li>If the degree of the numerator is greater than the degree of the denominator by\u00a0<em>more than one<\/em>, the end behavior of the function&#8217;s graph will mimic that of the graph of the reduced ratio of leading terms.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Case 3<\/strong>: Degree of numerator <em>is equal to<\/em> degree of denominator: horizontal asymptote at ratio of leading coefficients.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying Horizontal and Slant Asymptotes<\/h3>\n<p>For the functions below, identify the horizontal or slant asymptote.<\/p>\n<ol>\n<li>[latex]g\\left(x\\right)=\\dfrac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[\/latex]<\/li>\n<li>[latex]h\\left(x\\right)=\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex]<\/li>\n<li>[latex]k\\left(x\\right)=\\dfrac{{x}^{2}+4x}{{x}^{3}-8}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q968793\">Show Solution<\/span><\/p>\n<div id=\"q968793\" class=\"hidden-answer\" style=\"display: none\">\n<p>For these solutions, we will use [latex]f\\left(x\\right)=\\dfrac{p\\left(x\\right)}{q\\left(x\\right)}, q\\left(x\\right)\\ne 0[\/latex].<\/p>\n<ol>\n<li>[latex]g\\left(x\\right)=\\dfrac{6{x}^{3}-10x}{2{x}^{3}+5{x}^{2}}[\/latex]: The degree of [latex]p[\/latex] and the degree of [latex]q[\/latex] are both equal to 3, so we can find the horizontal asymptote by taking the ratio of the leading terms. There is a horizontal asymptote at [latex]y=\\frac{6}{2}[\/latex] or [latex]y=3[\/latex].<\/li>\n<li>[latex]h\\left(x\\right)=\\dfrac{{x}^{2}-4x+1}{x+2}[\/latex]: The degree of [latex]p=2[\/latex] and degree of [latex]q=1[\/latex]. Since [latex]p>q[\/latex] by 1, there is a slant asymptote. Perform polynomial long divsion to find the slant asympote. The slant asymptote is [latex]y=-x - 6[\/latex].<\/li>\n<li>[latex]k\\left(x\\right)=\\dfrac{{x}^{2}+4x}{{x}^{3}-8}[\/latex]: The degree of [latex]p=2\\text{ }<[\/latex] degree of [latex]q=3[\/latex], so there is a horizontal asymptote [latex]y=0[\/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=\"ohm221906\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=221906&theme=oea&iframe_resize_id=ohm221906&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video to see more worked examples of determining which kind of horizontal asymptote a rational function will have.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Determine Horizontal Asymptotes of Rational Functions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/A1tApZSE8nI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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-3895\">\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>Modification and Revision . <strong>Provided by<\/strong>: Lumen Learning. <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>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/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":17533,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision \",\"author\":\"\",\"organization\":\"Lumen 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