{"id":248743,"date":"2025-07-08T17:45:31","date_gmt":"2025-07-08T17:45:31","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=248743"},"modified":"2025-07-08T17:45:33","modified_gmt":"2025-07-08T17:45:33","slug":"find-the-vertical-horizontal-and-oblique-asymptotes-_-if-any-for-the-given-rational-function-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/08\/find-the-vertical-horizontal-and-oblique-asymptotes-_-if-any-for-the-given-rational-function-2\/","title":{"rendered":"Find the vertical, horizontal, and oblique asymptotes _ if any; for the given rational function"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Find the vertical, horizontal, and oblique asymptotes _ if any; for the given rational function: x4 16 G(x) = 2&#215;2 _ 4x Select the correct choice below and fill in any answer boxes within your choice. OA The vertical asymptote(s) islare x = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression ) 0B. There is no vertical asymptote_ Select the correct choice below and fill in any answer boxes within your choice_ OA The horizontal asymptote(s) islare y (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression ) 0B. There is no horizontal asymptote_ Select the correct choice below and fill in any answer boxes within your choice. OA The oblique asymptote(s) islare y (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression ) 0B. There is no oblique asymptote.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let&#8217;s analyze the rational function:G(x)=2&#215;2\u22124xx4+16G(x) = \\frac{2x^2 &#8211; 4x}{x^4 + 16}G(x)=x4+162&#215;2\u22124x\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Vertical Asymptotes<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">To find the vertical asymptotes, we look for values of xxx where the denominator is zero (and the numerator is non-zero). This happens when:x4+16=0x^4 + 16 = 0x4+16=0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">However, this equation does not have real solutions. The expression x4+16=0x^4 + 16 = 0x4+16=0 gives:x4=\u221216x^4 = -16&#215;4=\u221216<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This does not have real solutions, since a fourth power of a real number cannot be negative. Therefore, there are <strong>no vertical asymptotes<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer for vertical asymptotes<\/strong>:<br>There is no vertical asymptote.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Horizontal Asymptotes<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Next, we find the horizontal asymptote by analyzing the degrees of the numerator and denominator.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The degree of the numerator ( 2&#215;2\u22124x2x^2 &#8211; 4x2x2\u22124x ) is 2.<\/li>\n\n\n\n<li>The degree of the denominator ( x4+16x^4 + 16&#215;4+16 ) is 4.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">For rational functions, the horizontal asymptote is determined as follows:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0y = 0y=0.<\/li>\n\n\n\n<li>If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.<\/li>\n\n\n\n<li>If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Since the degree of the numerator (2) is less than the degree of the denominator (4), the horizontal asymptote is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer for horizontal asymptote<\/strong>:<br>y=0y = 0y=0<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Oblique Asymptotes<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">An oblique (slant) asymptote occurs when the degree of the numerator is exactly one higher than the degree of the denominator. In this case, the degree of the numerator (2) is not one higher than the degree of the denominator (4). Therefore, there is <strong>no oblique asymptote<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer for oblique asymptote<\/strong>:<br>There is no oblique asymptote.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Summary of answers:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Vertical asymptote(s): There is no vertical asymptote.<\/li>\n\n\n\n<li>Horizontal asymptote(s): y=0y = 0y=0.<\/li>\n\n\n\n<li>Oblique asymptote(s): There is no oblique asymptote.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner8-306.jpeg\" alt=\"\" class=\"wp-image-248744\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the vertical, horizontal, and oblique asymptotes _ if any; for the given rational function: x4 16 G(x) = 2&#215;2 _ 4x Select the correct choice below and fill in any answer boxes within your choice. OA The vertical asymptote(s) islare x = (Use a comma to separate answers as needed. Use integers or fractions [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-248743","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248743","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=248743"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248743\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=248743"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=248743"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=248743"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}