{"id":186313,"date":"2025-01-24T17:42:06","date_gmt":"2025-01-24T17:42:06","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=186313"},"modified":"2025-01-24T17:42:08","modified_gmt":"2025-01-24T17:42:08","slug":"describe-the-end-behavior-of-the-following-function-using-limit-notation-hx-x-5x2-19","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/24\/describe-the-end-behavior-of-the-following-function-using-limit-notation-hx-x-5x2-19\/","title":{"rendered":"Describe the end behavior of the following function using limit notation h(x)=-x&#8221; +5&#215;2-19"},"content":{"rendered":"\n<p>Describe the end behavior of the following function using limit notation h(x)=-x&#8221; +5&#215;2-19<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To describe the end behavior of the function ( h(x) = -x^3 + 5x^2 &#8211; 19 ), we analyze how the function behaves as ( x ) approaches positive and negative infinity. In other words, we want to find the limits of the function as ( x ) gets very large (positively and negatively).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">End Behavior:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>As ( x \\to \\infty ) (as ( x ) becomes very large):<\/strong> The function ( h(x) ) is a cubic polynomial, so we focus on the highest degree term for the behavior at large values of ( x ). The highest degree term in ( h(x) ) is ( -x^3 ), since the cubic term will dominate as ( x ) becomes large. [<br>\\lim_{x \\to \\infty} h(x) = \\lim_{x \\to \\infty} (-x^3 + 5x^2 &#8211; 19)<br>] As ( x \\to \\infty ), ( -x^3 ) becomes very negative, overpowering the other terms. So, [<br>\\lim_{x \\to \\infty} h(x) = -\\infty<br>]<\/li>\n\n\n\n<li><strong>As ( x \\to -\\infty ) (as ( x ) becomes very negative):<\/strong> Similarly, we consider the behavior as ( x \\to -\\infty ). The cubic term ( -x^3 ) still dominates, but since ( x^3 ) is negative when ( x ) is negative, ( -x^3 ) will become very positive. Therefore, the limit is: [<br>\\lim_{x \\to -\\infty} h(x) = \\lim_{x \\to -\\infty} (-x^3 + 5x^2 &#8211; 19)<br>] As ( x \\to -\\infty ), the term ( -x^3 ) grows large positively (since a negative raised to an odd power gives a negative, and multiplying by a negative flips it), and this causes the function to tend towards positive infinity. Thus, [<br>\\lim_{x \\to -\\infty} h(x) = \\infty<br>]<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The end behavior of the function is: [<br>\\lim_{x \\to \\infty} h(x) = -\\infty<br>]<br>[<br>\\lim_{x \\to -\\infty} h(x) = \\infty<br>]<\/li>\n<\/ul>\n\n\n\n<p>This means that as ( x ) increases without bound, the function decreases without bound (approaches negative infinity), and as ( x ) decreases without bound, the function increases without bound (approaches positive infinity).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Describe the end behavior of the following function using limit notation h(x)=-x&#8221; +5&#215;2-19 The Correct Answer and Explanation is : To describe the end behavior of the function ( h(x) = -x^3 + 5x^2 &#8211; 19 ), we analyze how the function behaves as ( x ) approaches positive and negative infinity. In other words, [&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-186313","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186313","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=186313"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186313\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=186313"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=186313"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=186313"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}