{"id":9936,"date":"2022-02-03T18:29:31","date_gmt":"2022-02-03T12:59:31","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9936"},"modified":"2022-02-03T18:31:43","modified_gmt":"2022-02-03T13:01:43","slug":"limits-at-infinity-definition-and-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/limits-at-infinity-definition-and-examples\/","title":{"rendered":"Limits at Infinity – Definition and Examples"},"content":{"rendered":"

Here you will learn how to solve or evaluate limits at infinity with examples.<\/p>\n

Let’s begin –<\/p>\n

Limits at Infinity<\/h2>\n

Algorithm to evaluate limits at infinity :<\/p>\n

1). Write down the given expression in the form of a rational function i.e. \\(f(x)\\over g(x)\\), if it is not so.<\/p>\n

2). If k is the highest power of x in numerator and denominator both, then divide each term in numerator and denominator by \\(x^k\\).<\/p>\n

3). Use the results \\(\\displaystyle{\\lim_{x \\to \\infty}}\\) \\(c\\over x^n\\) = 0 and \\(\\displaystyle{\\lim_{x \\to \\infty}}\\) c = c, where n > 0.<\/p><\/blockquote>\n

Also Read : How to Solve Indeterminate Forms of Limits<\/a><\/p>\n

Following examples will illustrate the above algorithm.<\/p>\n

Example<\/strong><\/span> : Evaluate \\(\\displaystyle{\\lim_{x \\to \\infty}}\\) \\(ax^2 + bx + c\\over dx^2 + ex + f\\).<\/p>\n

Solution<\/strong><\/span> : Here the expression assumes the form \\(\\infty\\over \\infty\\).<\/p>\n

We notice that the highest power of x in both the numerator and denominator is 2.<\/p>\n

So we divide each term in both the numerator and denominator by \\(x^2\\).<\/p>\n

\\(\\therefore\\)  \\(\\displaystyle{\\lim_{x \\to \\infty}}\\) \\(ax^2 + bx + c\\over dx^2 + ex + f\\)<\/p>\n

= \\(\\displaystyle{\\lim_{x \\to \\infty}}\\) \\(a + {b\\over x} + {c\\over x^2}\\over d + {e\\over x} + {f\\over x^2}\\)<\/p>\n

= \\(a + 0 + 0\\over d + 0 + 0\\) = \\(a\\over d\\)<\/p>\n

Example<\/strong><\/span> : Evaluate the limit : \\(\\displaystyle{\\lim_{x \\to \\infty}}\\) \\(x^2 + x + 1\\over {3x^2 + 2x – 5}\\).<\/p>\n

Solution<\/strong><\/span> : Here the expression assumes the form \\(\\infty\\over \\infty\\).<\/p>\n

We notice that the highest power of x in both the numerator and denominator is 2.<\/p>\n

So we divide each term in both the numerator and denominator by \\(x^2\\).<\/p>\n

\\(\\therefore\\) \\(\\displaystyle{\\lim_{x \\to \\infty}}\\)\\(x^2 + x + 1\\over {3x^2 + 2x – 5}\\)<\/p>\n

Limit = \\(\\displaystyle{\\lim_{x \\to 0}}\\) \\(1 + x + x^2\\over {3 + 2x – 5x^2}\\) = \\(1\\over 3\\)<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to solve or evaluate limits at infinity with examples. Let’s begin – Limits at Infinity Algorithm to evaluate limits at infinity : 1). Write down the given expression in the form of a rational function i.e. \\(f(x)\\over g(x)\\), if it is not so. 2). If k is the highest power …<\/p>\n

Limits at Infinity – Definition and Examples<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[19],"tags":[],"yoast_head":"\nLimits at Infinity - Definition and Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to solve or evaluate limits at infinity with definition and examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/limits-at-infinity-definition-and-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Limits at Infinity - 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