{"id":248702,"date":"2025-07-08T16:58:08","date_gmt":"2025-07-08T16:58:08","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=248702"},"modified":"2025-07-08T16:58:10","modified_gmt":"2025-07-08T16:58:10","slug":"find-the-vertical-horizontal-and-oblique-asymptotes-_-if-any-for-the-given-rational-function","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\/","title":{"rendered":"Find the vertical, horizontal, and oblique asymptotes _ if any; for the given rational function"},"content":{"rendered":"\n

Find the vertical, horizontal, and oblique asymptotes _ if any; for the given rational function: x4 16 G(x) = 2×2 _ 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

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To analyze the rational function G(x) = (x\u2074 + 16) \/ (2x\u00b2 – 4x)<\/strong> we will determine its vertical, horizontal, and oblique asymptotes.<\/p>\n\n\n\n

\ud83d\udd39 Vertical Asymptotes<\/h3>\n\n\n\n

Vertical asymptotes occur where the denominator is zero and the numerator is nonzero.<\/p>\n\n\n\n

Factor the denominator: 2x\u00b2 – 4x = 2x(x – 2)<\/p>\n\n\n\n

Set the denominator equal to zero: 2x(x – 2) = 0 x = 0 or x = 2<\/p>\n\n\n\n

Now check the numerator at these x-values: G(0) = (0\u2074 + 16) \/ (0) \u2192 undefined G(2) = (2\u2074 + 16) \/ (2(2 – 2)) = 32 \/ 0 \u2192 undefined<\/p>\n\n\n\n

Since the numerator is nonzero at both x = 0 and x = 2, these are vertical asymptotes.<\/p>\n\n\n\n

\u2705 Vertical asymptotes: x = 0, 2<\/strong><\/p>\n\n\n\n

\ud83d\udd39 Horizontal Asymptotes<\/h3>\n\n\n\n

Horizontal asymptotes depend on the degrees of the numerator and denominator.<\/p>\n\n\n\n

    \n
  • Degree of numerator: 4 (from x\u2074)<\/li>\n\n\n\n
  • Degree of denominator: 2 (from 2x\u00b2)<\/li>\n<\/ul>\n\n\n\n

    Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.<\/p>\n\n\n\n

    \u2705 Horizontal asymptote: There is no horizontal asymptote<\/strong><\/p>\n\n\n\n

    \ud83d\udd39 Oblique Asymptotes<\/h3>\n\n\n\n

    Oblique (slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator.<\/p>\n\n\n\n

    Here, the degree of the numerator is 4 and the denominator is 2. The difference is 2, not 1, so there is no oblique asymptote.<\/p>\n\n\n\n

    \u2705 Oblique asymptote: There is no oblique asymptote<\/strong><\/p>\n\n\n\n

    \u270d\ufe0f Summary<\/h3>\n\n\n\n
      \n
    • Vertical asymptotes: x = 0, 2<\/strong><\/li>\n\n\n\n
    • Horizontal asymptote: None<\/strong><\/li>\n\n\n\n
    • Oblique asymptote: None<\/strong><\/li>\n<\/ul>\n\n\n\n

      This function grows faster in the numerator than the denominator, so it diverges to infinity as x becomes large in magnitude. The vertical asymptotes represent values where the function is undefined due to division by zero. Since the numerator dominates the denominator by more than one degree, the function has neither a horizontal nor an oblique asymptote.<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      Find the vertical, horizontal, and oblique asymptotes _ if any; for the given rational function: x4 16 G(x) = 2×2 _ 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 […]<\/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-248702","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248702","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=248702"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/248702\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=248702"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=248702"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=248702"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}