{"id":190434,"date":"2025-02-12T09:21:11","date_gmt":"2025-02-12T09:21:11","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=190434"},"modified":"2025-02-12T09:21:13","modified_gmt":"2025-02-12T09:21:13","slug":"find-limit-lim_x-rightarrow-infinity-arctan","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/12\/find-limit-lim_x-rightarrow-infinity-arctan\/","title":{"rendered":"Find Limit Lim_x Rightarrow Infinity Arctan"},"content":{"rendered":"\n

Find Limit. Lim_x Rightarrow Infinity Arctan (X^2 + 1\/X + 1)<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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

It looks like the link you provided is not accessible directly through my tools. However, based on the problem you’ve mentioned \u2014 finding the limit of (\\arctan(x^2 + \\frac{1}{x} + 1)) as (x \\to \\infty), I can explain the solution.<\/p>\n\n\n\n

Problem:<\/h3>\n\n\n\n

Find the limit:
[
\\lim_{x \\to \\infty} \\arctan \\left( x^2 + \\frac{1}{x} + 1 \\right)
]<\/p>\n\n\n\n

Solution:<\/h3>\n\n\n\n
    \n
  1. Understand the behavior of the function:<\/strong>
    The given expression involves (\\arctan), which is a function with a horizontal asymptote at (\\frac{\\pi}{2}). Specifically:
    [
    \\lim_{y \\to \\infty} \\arctan(y) = \\frac{\\pi}{2}
    ]
    So, as the argument inside the (\\arctan) increases, the overall function approaches (\\frac{\\pi}{2}).<\/li>\n\n\n\n
  2. Simplify the expression inside (\\arctan):<\/strong>
    As (x \\to \\infty), the term (\\frac{1}{x}) approaches zero, and the expression inside the (\\arctan) becomes:
    [
    x^2 + \\frac{1}{x} + 1 \\approx x^2 + 1
    ]
    This is because (x^2) dominates the (\\frac{1}{x}) term as (x) grows larger.<\/li>\n\n\n\n
  3. Limit of the argument:<\/strong>
    As (x \\to \\infty), the expression (x^2 + 1) becomes infinitely large. Hence, we can substitute:
    [
    \\lim_{x \\to \\infty} \\left( x^2 + \\frac{1}{x} + 1 \\right) = \\infty
    ]<\/li>\n\n\n\n
  4. Final Limit:<\/strong>
    Since the argument of the (\\arctan) function approaches infinity as (x \\to \\infty), we can apply the limit property of (\\arctan):
    [
    \\lim_{x \\to \\infty} \\arctan \\left( x^2 + \\frac{1}{x} + 1 \\right) = \\arctan(\\infty) = \\frac{\\pi}{2}
    ]<\/li>\n<\/ol>\n\n\n\n

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

    The limit of (\\arctan(x^2 + \\frac{1}{x} + 1)) as (x \\to \\infty) is (\\frac{\\pi}{2}).<\/p>\n\n\n\n

    This solution shows how the asymptotic behavior of the function simplifies the limit problem, and understanding the asymptote of the arctangent function is key to determining the final answer.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Find Limit. Lim_x Rightarrow Infinity Arctan (X^2 + 1\/X + 1) The Correct Answer and Explanation is : It looks like the link you provided is not accessible directly through my tools. However, based on the problem you’ve mentioned \u2014 finding the limit of (\\arctan(x^2 + \\frac{1}{x} + 1)) as (x \\to \\infty), I can […]<\/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-190434","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190434","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=190434"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190434\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=190434"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=190434"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=190434"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}