{"id":1024,"date":"2025-07-21T22:15:05","date_gmt":"2025-07-21T22:15:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1024"},"modified":"2026-01-14T18:28:01","modified_gmt":"2026-01-14T18:28:01","slug":"rational-functions-learn-it-2-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rational-functions-learn-it-2-2\/","title":{"raw":"Rational Functions: Learn It 2","rendered":"Rational Functions: Learn It 2"},"content":{"raw":"<h2>Local Behavior of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/h2>\r\nConsider the function [latex]f(x) = \\dfrac{1}{x}[\/latex].\r\n\r\n<img class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213912\/CNX_Precalc_Figure_03_07_0032.jpg\" alt=\"Graph of f(x)=1\/x with its vertical asymptote at x=0.\" width=\"400\" height=\"299\" \/>\r\n\r\nPreviously, we stated that:\r\n<ul>\r\n \t<li>As [latex]x\\to {0}^{-}, f\\left(x\\right)\\to -\\infty [\/latex]<\/li>\r\n \t<li>As [latex]x\\to {0}^{+}, f\\left(x\\right)\\to \\infty [\/latex].<\/li>\r\n<\/ul>\r\nThis behavior creates a <strong>vertical asymptote<\/strong>, which is a vertical line that the graph approaches but never crosses. In this case, as the input nears zero from the left, the function value decreases without bound. As the input nears zero from the right, the function value increases without bound. The line\u00a0[latex]x=0[\/latex] is a vertical asymptote for the function.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>vertical asymptote<\/h3>\r\nA <strong>vertical asymptote<\/strong> of a graph is a vertical line [latex]x=a[\/latex] where the graph tends toward positive or negative infinity as the inputs approach [latex]x[\/latex].\r\n\r\nWe write:\r\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to a,f\\left(x\\right)\\to \\infty , \\text{or as }x\\to a,f\\left(x\\right)\\to -\\infty [\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox connectIt\" aria-label=\"Connect It\">The term\u00a0<em>asymptote<\/em> has its origins from three Greek roots.\r\n<p style=\"padding-left: 30px;\"><em>a<\/em>, meaning\u00a0<em>not<\/em><\/p>\r\n<p style=\"padding-left: 30px;\"><em>sym (or syn),<\/em> meaning\u00a0<em>together<\/em><\/p>\r\n<p style=\"padding-left: 30px;\"><em>ptotos,<\/em> meaning\u00a0<em>to fall<\/em><\/p>\r\nThese give us <em>asymptotos,\u00a0<\/em>meaning\u00a0<em>not falling together,\u00a0<\/em>which leads to the modern term describing a line that a curve (the graph of a function) approaches but never meets.\r\n\r\n<\/section>\r\n<h2>End Behavior of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/h2>\r\nLet's take a look at the end behavior of the function [latex]f(x) = \\dfrac{1}{x}[\/latex] again.<img class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213915\/CNX_Precalc_Figure_03_07_0052.jpg\" alt=\"Graph of f(x)=1\/x with its vertical asymptote at x=0 and its horizontal asymptote at y=0.\" width=\"400\" height=\"299\" \/>\r\n\r\nAs the values of [latex]x[\/latex]\u00a0approach infinity, the function values approach [latex]0[\/latex]. As the values of [latex]x[\/latex]\u00a0approach negative infinity, the function values approach [latex]0[\/latex]. Symbolically, using arrow notation\r\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to \\infty ,f\\left(x\\right)\\to 0,\\text{and as }x\\to -\\infty ,f\\left(x\\right)\\to 0[\/latex].<\/p>\r\nBased on this overall behavior and the graph, we can see that the function approaches [latex]0[\/latex] but never actually reaches [latex]0[\/latex]; it seems to level off as the inputs become large. This behavior creates a <strong>horizontal asymptote<\/strong>, a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line [latex]y=0[\/latex].\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>horizontal asymptote<\/h3>\r\nA <strong>horizontal asymptote<\/strong> of a graph is a horizontal line [latex]y=b[\/latex] where the graph approaches the line as the inputs increase or decrease without bound.\r\n\r\nWe write\r\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to \\infty \\text{ or }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to b[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Use arrow notation to describe the end behavior and local behavior of the function below.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213918\/CNX_Precalc_Figure_03_07_0062.jpg\" alt=\"Graph of f(x)=1\/(x-2)+4 with its vertical asymptote at x=2 and its horizontal asymptote at y=4.\" width=\"487\" height=\"477\" \/>[reveal-answer q=\"444547\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"444547\"]Notice that the graph is showing a vertical asymptote at [latex]x=2[\/latex], which tells us that the function is undefined at [latex]x=2[\/latex].\r\n<p style=\"text-align: center;\">As [latex]x\\to {2}^{-},\\hspace{2mm}f\\left(x\\right)\\to -\\infty[\/latex], and as [latex]x\\to {2}^{+},\\text{ }f\\left(x\\right)\\to \\infty [\/latex]<\/p>\r\nAnd as the inputs decrease without bound, the graph appears to be leveling off at output values of 4, indicating a horizontal asymptote at [latex]y=4[\/latex]. As the inputs increase without bound, the graph levels off at 4.\r\n<p style=\"text-align: center;\">As [latex]x\\to \\infty ,\\text{ }f\\left(x\\right)\\to 4[\/latex], and as [latex]x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to 4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]318941[\/ohm_question]<\/section>","rendered":"<h2>Local Behavior of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/h2>\n<p>Consider the function [latex]f(x) = \\dfrac{1}{x}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213912\/CNX_Precalc_Figure_03_07_0032.jpg\" alt=\"Graph of f(x)=1\/x with its vertical asymptote at x=0.\" width=\"400\" height=\"299\" \/><\/p>\n<p>Previously, we stated that:<\/p>\n<ul>\n<li>As [latex]x\\to {0}^{-}, f\\left(x\\right)\\to -\\infty[\/latex]<\/li>\n<li>As [latex]x\\to {0}^{+}, f\\left(x\\right)\\to \\infty[\/latex].<\/li>\n<\/ul>\n<p>This behavior creates a <strong>vertical asymptote<\/strong>, which is a vertical line that the graph approaches but never crosses. In this case, as the input nears zero from the left, the function value decreases without bound. As the input nears zero from the right, the function value increases without bound. The line\u00a0[latex]x=0[\/latex] is a vertical asymptote for the function.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>vertical asymptote<\/h3>\n<p>A <strong>vertical asymptote<\/strong> of a graph is a vertical line [latex]x=a[\/latex] where the graph tends toward positive or negative infinity as the inputs approach [latex]x[\/latex].<\/p>\n<p>We write:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to a,f\\left(x\\right)\\to \\infty , \\text{or as }x\\to a,f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n<\/section>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">The term\u00a0<em>asymptote<\/em> has its origins from three Greek roots.<\/p>\n<p style=\"padding-left: 30px;\"><em>a<\/em>, meaning\u00a0<em>not<\/em><\/p>\n<p style=\"padding-left: 30px;\"><em>sym (or syn),<\/em> meaning\u00a0<em>together<\/em><\/p>\n<p style=\"padding-left: 30px;\"><em>ptotos,<\/em> meaning\u00a0<em>to fall<\/em><\/p>\n<p>These give us <em>asymptotos,\u00a0<\/em>meaning\u00a0<em>not falling together,\u00a0<\/em>which leads to the modern term describing a line that a curve (the graph of a function) approaches but never meets.<\/p>\n<\/section>\n<h2>End Behavior of [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/h2>\n<p>Let&#8217;s take a look at the end behavior of the function [latex]f(x) = \\dfrac{1}{x}[\/latex] again.<img loading=\"lazy\" decoding=\"async\" class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213915\/CNX_Precalc_Figure_03_07_0052.jpg\" alt=\"Graph of f(x)=1\/x with its vertical asymptote at x=0 and its horizontal asymptote at y=0.\" width=\"400\" height=\"299\" \/><\/p>\n<p>As the values of [latex]x[\/latex]\u00a0approach infinity, the function values approach [latex]0[\/latex]. As the values of [latex]x[\/latex]\u00a0approach negative infinity, the function values approach [latex]0[\/latex]. Symbolically, using arrow notation<\/p>\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to \\infty ,f\\left(x\\right)\\to 0,\\text{and as }x\\to -\\infty ,f\\left(x\\right)\\to 0[\/latex].<\/p>\n<p>Based on this overall behavior and the graph, we can see that the function approaches [latex]0[\/latex] but never actually reaches [latex]0[\/latex]; it seems to level off as the inputs become large. This behavior creates a <strong>horizontal asymptote<\/strong>, a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line [latex]y=0[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>horizontal asymptote<\/h3>\n<p>A <strong>horizontal asymptote<\/strong> of a graph is a horizontal line [latex]y=b[\/latex] where the graph approaches the line as the inputs increase or decrease without bound.<\/p>\n<p>We write<\/p>\n<p style=\"text-align: center;\">[latex]\\text{As }x\\to \\infty \\text{ or }x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to b[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Use arrow notation to describe the end behavior and local behavior of the function below.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02213918\/CNX_Precalc_Figure_03_07_0062.jpg\" alt=\"Graph of f(x)=1\/(x-2)+4 with its vertical asymptote at x=2 and its horizontal asymptote at y=4.\" width=\"487\" height=\"477\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q444547\">Show Solution<\/button><\/p>\n<div id=\"q444547\" class=\"hidden-answer\" style=\"display: none\">Notice that the graph is showing a vertical asymptote at [latex]x=2[\/latex], which tells us that the function is undefined at [latex]x=2[\/latex].<\/p>\n<p style=\"text-align: center;\">As [latex]x\\to {2}^{-},\\hspace{2mm}f\\left(x\\right)\\to -\\infty[\/latex], and as [latex]x\\to {2}^{+},\\text{ }f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<p>And as the inputs decrease without bound, the graph appears to be leveling off at output values of 4, indicating a horizontal asymptote at [latex]y=4[\/latex]. As the inputs increase without bound, the graph levels off at 4.<\/p>\n<p style=\"text-align: center;\">As [latex]x\\to \\infty ,\\text{ }f\\left(x\\right)\\to 4[\/latex], and as [latex]x\\to -\\infty ,\\text{ }f\\left(x\\right)\\to 4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm318941\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318941&theme=lumen&iframe_resize_id=ohm318941&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":508,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1024"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1024\/revisions"}],"predecessor-version":[{"id":5350,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1024\/revisions\/5350"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/508"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1024\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1024"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1024"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1024"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1024"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}