{"id":237979,"date":"2025-06-17T18:59:43","date_gmt":"2025-06-17T18:59:43","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=237979"},"modified":"2025-06-17T18:59:45","modified_gmt":"2025-06-17T18:59:45","slug":"the-graph-of-which-of-the-following-functions-has-exactly-one-horizontal-asymptote-and-no-vertical-asymptotes","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/17\/the-graph-of-which-of-the-following-functions-has-exactly-one-horizontal-asymptote-and-no-vertical-asymptotes\/","title":{"rendered":"The graph of which of the following functions has exactly one horizontal asymptote and no vertical asymptotes"},"content":{"rendered":"\n

)<\/p>\n\n\n\n

    \n
  1. The graph of which of the following functions has exactly one horizontal asymptote and no vertical asymptotes?
    A) y = (1)\/(x^2 + 1)
    B) y = (1)\/(x^3 + 1)
    C) y = (1)\/(e^x + 1)
    D) y = (1)\/(x^2 – 1)
    Added by<\/li>\n<\/ol>\n\n\n\n
    \"\"<\/figure>\n\n\n\n

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

    The correct answer is C) y = (1)\/(e^x + 1)<\/strong>.<\/p>\n\n\n\n

    Explanation:<\/h3>\n\n\n\n

    To determine which function has exactly one horizontal asymptote and no vertical asymptotes<\/strong>, we analyze each option:<\/p>\n\n\n\n

    1. Horizontal Asymptotes:<\/h4>\n\n\n\n

    A function has a horizontal asymptote when the limit of f(x)f(x) as xx approaches infinity or negative infinity tends toward a specific finite value.<\/p>\n\n\n\n

      \n
    • Option A: <\/strong>y=1×2+1y = \\frac{1}{x^2 + 1}\n
        \n
      • As x\u2192\u00b1\u221ex \\to \\pm\\infty, x2x^2 grows indefinitely, meaning x2+1\u2192\u221ex^2 + 1 \\to \\infty, and thus y\u21920y \\to 0.<\/li>\n\n\n\n
      • There is one horizontal asymptote<\/strong> at y=0y = 0.<\/li>\n\n\n\n
      • However, this function has no vertical asymptotes<\/strong> because the denominator never equals zero.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • Option B: <\/strong>y=1×3+1y = \\frac{1}{x^3 + 1}\n
          \n
        • For large xx, x3+1x^3 + 1 dominates, leading to y\u21920y \\to 0.<\/li>\n\n\n\n
        • One horizontal asymptote<\/strong> exists at y=0y = 0, and there are no vertical asymptotes<\/strong> since the denominator never vanishes.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
        • Option C: <\/strong>y=1ex+1y = \\frac{1}{e^x + 1}\n
            \n
          • As x\u2192\u221ex \\to \\infty, exe^x grows exponentially, meaning ex+1\u2192\u221ee^x + 1 \\to \\infty and thus y\u21920y \\to 0.<\/li>\n\n\n\n
          • As x\u2192\u2212\u221ex \\to -\\infty, exe^x approaches 0, making y=11=1y = \\frac{1}{1} = 1.<\/li>\n\n\n\n
          • Two horizontal asymptotes<\/strong> exist at y=0y = 0 and y=1y = 1, eliminating this option<\/strong>.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
          • Option D: <\/strong>y=1×2\u22121y = \\frac{1}{x^2 – 1}\n
              \n
            • The denominator equals zero when x2=1x^2 = 1, meaning vertical asymptotes exist at x=\u00b11x = \\pm1.<\/li>\n\n\n\n
            • This contradicts the given condition.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n

              Thus, Option C is the correct answer<\/strong> because it has exactly one horizontal asymptote<\/strong> and no vertical asymptotes<\/strong>.<\/p>\n\n\n\n

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

              ) The Correct Answer and Explanation is: The correct answer is C) y = (1)\/(e^x + 1). Explanation: To determine which function has exactly one horizontal asymptote and no vertical asymptotes, we analyze each option: 1. Horizontal Asymptotes: A function has a horizontal asymptote when the limit of f(x)f(x) as xx approaches infinity or negative […]<\/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-237979","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237979","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=237979"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237979\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=237979"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=237979"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=237979"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}