{"id":4254,"date":"2022-06-10T19:01:41","date_gmt":"2022-06-10T19:01:41","guid":{"rendered":"https:\/\/mdr.foobrdigital.com\/?p=4254"},"modified":"2022-06-10T19:01:41","modified_gmt":"2022-06-10T19:01:41","slug":"horizontal-asymptote-of-a-rational-function","status":"publish","type":"post","link":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/2022\/06\/10\/horizontal-asymptote-of-a-rational-function\/","title":{"rendered":"Horizontal Asymptote of a Rational Function"},"content":{"rendered":"\n<p>A\u00a0horizontal asymptote\u00a0(HA) of a function is an imaginary horizontal line to which its graph appears to be very close but never touch. It is of the form y = some number. Here, &#8220;some number&#8221; is closely connected to the excluded values from the range. A rational function can have at most one horizontal asymptote. Easy way to find the horizontal asymptote of a rational function is using the degrees of the numerator (N) and denominators (D).<\/p>\n\n\n\n<ul><li>If N &lt; D, then there is a HA at y = 0.<\/li><li>If N > D, then there is no HA.<\/li><li>If N = D, then the HA is y =\u00a0ratio\u00a0of the leading coefficients.<\/li><\/ul>\n\n\n\n<p><strong>Example:<\/strong>&nbsp;Find the horizontal asymptote (if any) of the function f(x) = (x<sup>2<\/sup>&nbsp;+ 5x + 6) \/ (x<sup>2<\/sup>&nbsp;+ x &#8211; 2).<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Here the degree of the numerator is, N = 2, and the degree of the denominator is, D = 2.<\/p>\n\n\n\n<p>Since N = D, the HA is y = (leading coefficient of numerator) \/ (leading coefficient of denominator) = 1\/1 = 1.<\/p>\n\n\n\n<p>Thus, the HA is y = 1.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Slant (Oblique) Asymptotes of a Rational Function<\/h3>\n\n\n\n<p>A slant asymptote is also an imaginary oblique line to which a part of the graph appears to touch. A rational function has a slant asymptote only when the degree of the numerator (N) is exactly one greater than the degree of the denominator (D). Its equation is y = quotient that is obtained by dividing the numerator by denominator using the\u00a0long division.<\/p>\n\n\n\n<p><strong>Example:<\/strong>&nbsp;Find the slant asymptote of the function f(x) = x<sup>2<\/sup>\/(x+1).<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Here the degree of numerator is 2 and that of denominator = 1. So it has a slant asymptote.<\/p>\n\n\n\n<p>Let us divide x<sup>2<\/sup>&nbsp;by (x + 1) by long division (or we can use synthetic division as well).<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/slant-asymptote-of-a-rational-function-1642146920.png\" alt=\"slant asymptote of a rational function using long division\"\/><\/figure>\n\n\n\n<p>Thus, the slant asymptote is y = x &#8211; 1.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A\u00a0horizontal asymptote\u00a0(HA) of a function is an imaginary horizontal line to which its graph appears to be very close but never touch. It is of the form y = some number. Here, &#8220;some number&#8221; is closely connected to the excluded values from the range. A rational function can have at most one horizontal asymptote. Easy [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[715],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/4254"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=4254"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/4254\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=4254"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=4254"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=4254"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}