{"id":266737,"date":"2025-07-23T07:00:03","date_gmt":"2025-07-23T07:00:03","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=266737"},"modified":"2025-07-23T07:00:07","modified_gmt":"2025-07-23T07:00:07","slug":"determine-the-limit-of-the-sequence-or-state-that-the-sequence-diverges","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/23\/determine-the-limit-of-the-sequence-or-state-that-the-sequence-diverges\/","title":{"rendered":"Determine the limit of the sequence or state that the sequence diverges."},"content":{"rendered":"\n

Determine the limit of the sequence or state that the sequence diverges. a_(n)=(n)\/(\\sqrt(n^(3)+13)) (Use symbolic notation and fractions where needed. Enter DNE if the sequence diverges.) \\lim_(n->\\infty )a_(n)=1.0 Incorrect Answer<\/p>\n\n\n\n

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

To determine the limit of the sequence an=nn3+13a_n = \\frac{n}{\\sqrt{n^3 + 13}}an\u200b=n3+13\u200bn\u200b, we need to analyze the behavior of the sequence as n\u2192\u221en \\to \\inftyn\u2192\u221e.<\/p>\n\n\n\n

Step-by-step solution:<\/h3>\n\n\n\n
    \n
  1. Express the sequence<\/strong>:
    We are given the sequence: an=nn3+13a_n = \\frac{n}{\\sqrt{n^3 + 13}}an\u200b=n3+13\u200bn\u200b<\/li>\n\n\n\n
  2. Factor out the dominant term<\/strong>:
    As nnn grows large, the term n3n^3n3 will dominate over the constant 13 inside the square root. To better understand the limit, factor n3n^3n3 out of the square root: an=nn3(1+13n3)a_n = \\frac{n}{\\sqrt{n^3 (1 + \\frac{13}{n^3})}}an\u200b=n3(1+n313\u200b)\u200bn\u200b Simplifying further: an=nn3\/21+13n3a_n = \\frac{n}{n^{3\/2} \\sqrt{1 + \\frac{13}{n^3}}}an\u200b=n3\/21+n313\u200b\u200bn\u200b Now, we can simplify the expression: an=1n1\/21+13n3a_n = \\frac{1}{n^{1\/2} \\sqrt{1 + \\frac{13}{n^3}}}an\u200b=n1\/21+n313\u200b\u200b1\u200b<\/li>\n\n\n\n
  3. Take the limit<\/strong> as n\u2192\u221en \\to \\inftyn\u2192\u221e:
    • As n\u2192\u221en \\to \\inftyn\u2192\u221e, the term 13n3\\frac{13}{n^3}n313\u200b tends to 0. Therefore, the square root term 1+13n3\\sqrt{1 + \\frac{13}{n^3}}1+n313\u200b\u200b approaches 1=1\\sqrt{1} = 11\u200b=1.<\/li>
    • Thus, the expression simplifies to:<\/li><\/ul>an\u22481n1\/2a_n \\approx \\frac{1}{n^{1\/2}}an\u200b\u2248n1\/21\u200b\n
        \n
      • As n\u2192\u221en \\to \\inftyn\u2192\u221e, 1n1\/2\\frac{1}{n^{1\/2}}n1\/21\u200b tends to 0.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n

        Conclusion:<\/h3>\n\n\n\n

        Therefore, the limit of the sequence is: lim\u2061n\u2192\u221ean=0\\lim_{n \\to \\infty} a_n = 0n\u2192\u221elim\u200ban\u200b=0<\/p>\n\n\n\n

        This means that the sequence converges to 0 as n\u2192\u221en \\to \\inftyn\u2192\u221e. The answer is 0<\/strong>.<\/p>\n\n\n\n

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

        Determine the limit of the sequence or state that the sequence diverges. a_(n)=(n)\/(\\sqrt(n^(3)+13)) (Use symbolic notation and fractions where needed. Enter DNE if the sequence diverges.) \\lim_(n->\\infty )a_(n)=1.0 Incorrect Answer The Correct Answer and Explanation is: To determine the limit of the sequence an=nn3+13a_n = \\frac{n}{\\sqrt{n^3 + 13}}an\u200b=n3+13\u200bn\u200b, we need to analyze the behavior of […]<\/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-266737","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266737","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=266737"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/266737\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=266737"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=266737"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=266737"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}