{"id":22,"date":"2023-09-17T11:07:37","date_gmt":"2023-09-17T11:07:37","guid":{"rendered":"https:\/\/mathority.org\/pt\/assintota-obliqua\/"},"modified":"2023-09-17T11:07:37","modified_gmt":"2023-09-17T11:07:37","slug":"assintota-obliqua","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/assintota-obliqua\/","title":{"rendered":"Ass\u00edntota obl\u00edqua"},"content":{"rendered":"<p>Neste artigo explicamos o que s\u00e3o as ass\u00edntotas obl\u00edquas de uma fun\u00e7\u00e3o. Voc\u00ea aprender\u00e1 quando uma fun\u00e7\u00e3o tem uma ass\u00edntota obl\u00edqua e como ela \u00e9 calculada. E, al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos de ass\u00edntotas obl\u00edquas e praticar com exerc\u00edcios resolvidos passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-una-asintota-oblicua\"><\/span> O que \u00e9 uma ass\u00edntota obl\u00edqua?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>A ass\u00edntota obl\u00edqua de uma fun\u00e7\u00e3o \u00e9 uma reta inclinada da qual seu gr\u00e1fico se aproxima indefinidamente, sem nunca cruz\u00e1-la.<\/strong> Consequentemente, todas as ass\u00edntotas obl\u00edquas s\u00e3o retas com a equa\u00e7\u00e3o <em>y=mx+n<\/em> .<\/p>\n<p> A inclina\u00e7\u00e3o e a origem de uma ass\u00edntota obl\u00edqua s\u00e3o calculadas usando as seguintes f\u00f3rmulas: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/asymptote-oblique-dune-fonction.webp\" alt=\"ass\u00edntota obl\u00edqua de uma fun\u00e7\u00e3o\" class=\"wp-image-1362\" width=\"290\" height=\"328\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-calcular-la-asintota-oblicua-de-una-funcion\"><\/span> Como calcular a ass\u00edntota obl\u00edqua de uma fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para calcular a ass\u00edntota obl\u00edqua de uma fun\u00e7\u00e3o, devem ser executados os seguintes passos:<\/p>\n<ol style=\"color:#FF8A05; font-weight: bold;border:\">\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\">Calcule o limite ao infinito da fun\u00e7\u00e3o dividida por x.<\/span><\/li>\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">Se o limite acima resultar em um n\u00famero real diferente de zero, significa que a fun\u00e7\u00e3o tem uma ass\u00edntota obl\u00edqua. E mais, a inclina\u00e7\u00e3o da referida ass\u00edntota obl\u00edqua ser\u00e1 o valor obtido no limite.<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-004f6e72e10d1ba23da76d2fd8ea13f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to \\pm\\infty}\\frac{f(x)}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"125\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">Neste caso, resta calcular a intercepta\u00e7\u00e3o da ass\u00edntota obl\u00edqua resolvendo o seguinte limite:<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9cc74ce0447b0a9148cae947674ad085_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} [f(x)-mx]\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"170\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/ol>\n<p> <strong>Nota:<\/strong> os limites devem ser calculados em mais e menos infinito, mas normalmente d\u00e3o o mesmo resultado e por isso simplificamos colocando \u00b1\u221e. Mas se os limites em mais e menos infinito fossem diferentes, a ass\u00edntota obl\u00edqua esquerda e a ass\u00edntota obl\u00edqua direita teriam de ser calculadas separadamente. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-asintota-oblicua\"><\/span> Exemplo de ass\u00edntota obl\u00edqua<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A seguir, pegaremos a ass\u00edntota obl\u00edqua da seguinte fun\u00e7\u00e3o racional para que voc\u00ea possa ver um exemplo de como isso \u00e9 feito:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b02f6283fd481e890a943badfa2c876f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^2+1}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"109\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> As ass\u00edntotas obl\u00edquas s\u00e3o do tipo<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ad313410fc976bc53709807aa8aed8e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=mx+n.\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -4px;\"><\/p>\n<p> Ent\u00e3o, primeiro calculamos a inclina\u00e7\u00e3o da reta<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> com sua f\u00f3rmula correspondente:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9b50ee5cbc3cf33f7fd42c3fe03a3d71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to \\pm\\infty} \\frac{f(x)}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"125\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-870fc158a1aabb54cb5f3b4296381512_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m= \\lim_{x \\to \\pm\\infty} \\cfrac{\\cfrac{x^2+1}{x}}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"141\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Para resolver este limite devemos aplicar as propriedades das fra\u00e7\u00f5es:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7f313d826cd1a2dd1ef66b1d0a40efb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\cfrac{a}{b}}{\\cfrac{c}{d}}=\\cfrac{a\\cdot d}{b\\cdot c}\" title=\"Rendered by QuickLaTeX.com\" height=\"80\" width=\"69\" style=\"vertical-align: -39px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dfc6b6aa917846535c6c4b6158961988_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m= \\lim_{x \\to \\pm\\infty} \\cfrac{\\cfrac{x^2+1}{x}}{x}=\\lim_{x \\to \\pm\\infty} \\cfrac{x^2+1}{x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"264\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> E agora calculamos o limite:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-542fc353481ddc465b7a40f665d3661d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to \\pm\\infty} \\cfrac{x^2+1}{x^2} = \\cfrac{+\\infty}{+\\infty} = \\cfrac{1}{1} = \\bm{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"269\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Nesse caso, o resultado da indetermina\u00e7\u00e3o do infinito entre o infinito \u00e9 a divis\u00e3o dos coeficientes de x de maior grau, pois o numerador e o denominador s\u00e3o da mesma ordem.<\/p>\n<p> O limite acima fornece um n\u00famero real diferente de zero, ent\u00e3o a fun\u00e7\u00e3o tem uma ass\u00edntota obl\u00edqua. Vamos agora calcular a intercepta\u00e7\u00e3o y<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> da ass\u00edntota usando sua f\u00f3rmula correspondente:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-45119c7a74d77a92d7a6cfd5b5c3544f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[f(x)-mx\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"173\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9197669cc0e41aa22224b552b21b31ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[\\cfrac{x^2+1}{x}-1x\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"191\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Tentamos calcular o limite:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4d7fa012eace37e82c243012c91f1a5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[\\cfrac{x^2+1}{x}-x\\right] = \\cfrac{+\\infty}{+\\infty} - (+\\infty) = \\bm{+\\infty - \\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"412\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Mas obtemos indetermina\u00e7\u00e3o infinito menos infinito. \u00c9, portanto, necess\u00e1rio reduzir os termos a um denominador comum. Para fazer isso, multiplicamos e dividimos x pelo denominador da fra\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a2355ed9411470b9fd20a50ebbd48726_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n=\\lim_{x \\to \\pm\\infty} \\left[\\cfrac{x^2+1}{x}-\\cfrac{x\\cdot x}{x} \\right] = \\lim_{x \\to \\pm\\infty} \\left[\\cfrac{x^2+1}{x}-\\cfrac{x^2}{x}\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"391\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Agora que os dois termos t\u00eam o mesmo denominador, podemos agrup\u00e1-los:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f932ebc8728669c7c6b57e115c444fc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[\\cfrac{x^2+1}{x}-\\cfrac{x^2}{x} \\right] =  \\lim_{x \\to \\pm\\infty} \\cfrac{x^2+1-x^2}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"358\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Operamos no numerador:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c39259f829c9e99fc88819c6ae266e82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty}  \\cfrac{\\phantom{2}1\\phantom{2}}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"112\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> E finalmente, resolvemos o limite:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16a0044416d02e77b05f65f1bb93d4cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty}  \\cfrac{\\phantom{2}1\\phantom{2}}{x}= \\cfrac{1}{\\pm\\infty} = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"201\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o <em>n<\/em> =0. Portanto, a ass\u00edntota obl\u00edqua \u00e9 uma fun\u00e7\u00e3o linear: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6fbe1cc5f3362ddbd80ed0b29c0bb4ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a68ac5c51acd0f68bd022aee64cd9cd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 1x+0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4909df7491ef54f0df1e922bc29417f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y=x}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> A fun\u00e7\u00e3o estudada est\u00e1 representada no gr\u00e1fico abaixo. Como voc\u00ea pode ver, a fun\u00e7\u00e3o chega muito perto da reta y=x mas nunca a toca porque \u00e9 uma ass\u00edntota obl\u00edqua: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/image-1.png\" alt=\"exemplo de ass\u00edntota obl\u00edqua\" class=\"wp-image-1374\" width=\"424\" height=\"478\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-asintotas-oblicuas\"><\/span> Exerc\u00edcios resolvidos sobre ass\u00edntotas obl\u00edquas<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Encontre a ass\u00edntota obl\u00edqua da seguinte fun\u00e7\u00e3o racional: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8ecc70adc78bf259cf6e36c0dcf1bee7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\frac{x^2+2x+3}{x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"150\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> As ass\u00edntotas obl\u00edquas t\u00eam a forma<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e4adcc4368f6296906b6231bf17a6a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<p> , \u00e9 portanto necess\u00e1rio calcular os par\u00e2metros <em>m<\/em> e <em>n<\/em> . Primeiro calculamos <em>m<\/em> aplicando sua f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc38f695cee95c4c60c6e2591345119e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to \\pm\\infty} \\frac{f(x)}{x} = \\lim_{x \\to \\pm\\infty} \\cfrac{\\cfrac{x^2+2x+3}{x+1}}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"293\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Simplificamos a fra\u00e7\u00e3o aplicando as propriedades das fra\u00e7\u00f5es: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef59ac0cd51c39c615896543993c12b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to \\pm\\infty} \\frac{x^2+2x+3}{(x+1)\\cdot x}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"180\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c51e373fd07a821f8e75d63e38f252dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to \\pm\\infty} \\frac{x^2+2x+3}{x^2+x}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"180\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E resolvemos o limite:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-653fa714bca94b5cc4f3ed715d7c1520_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to \\pm\\infty} \\frac{x^2+2x+3}{x^2+x}= \\frac{+\\infty}{+\\infty} = \\frac{1}{1} = \\bm{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"308\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o <em>m<\/em> = 1. Vamos agora calcular a intercepta\u00e7\u00e3o da ass\u00edntota obl\u00edqua aplicando sua f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e779b5ac239ae56c53427510dbd54dcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[f(x)-mx\\right] = \\lim_{x \\to \\pm\\infty} \\left[ \\frac{x^2+2x+3}{x+1}-1x\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"395\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Tentamos calcular o limite:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f95f290fbf258d45aa5765008d7aad13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[ \\frac{x^2+2x+3}{x+1}-x\\right]= \\bm{+\\infty - \\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"320\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mas obtemos a forma indeterminada infinito menos infinito. Devemos, portanto, reduzir os termos a um denominador comum e depois agrup\u00e1-los:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0712d34ed442d9e12ef2490f04df078a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[ \\frac{x^2+2x+3}{x+1}-x\\right] =\\\\[6ex]=\\displaystyle\\lim_{x \\to \\pm\\infty} \\left[ \\frac{x^2+2x+3}{x+1}-\\frac{x \\cdot (x+1)}{x+1} \\right] = \\\\[6ex]=\\displaystyle\\lim_{x \\to \\pm\\infty} \\left[ \\frac{x^2+2x+3}{x+1}-\\frac{x^2+x}{x+1} \\right]=\\\\[6ex]=\\displaystyle\\lim_{x \\to \\pm\\infty} \\frac{x^2+2x+3-(x^2+x)}{x+1}\\\\[6ex]\\displaystyle =\\lim_{x \\to \\pm\\infty} \\frac{x^2+2x+3-x^2-x}{x+1}=\\\\[6ex]=\\displaystyle \\lim_{x \\to \\pm\\infty} \\frac{x+3}{x+1}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"434\" width=\"300\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E finalmente, resolvemos o limite:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ee7e1fdd8e781abed322fed1182ddb15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n =\\lim_{x \\to \\pm\\infty} \\frac{x+3}{x+1} = \\frac{\\infty}{\\infty} = \\frac{1}{1} = \\bm{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"241\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resumindo, a ass\u00edntota obl\u00edqua da fun\u00e7\u00e3o \u00e9: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6fbe1cc5f3362ddbd80ed0b29c0bb4ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-69c0f50795c1f6034c0cd04201f614d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 1x + 1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5ffe94db5ae8fa1abc72e6007c2c0586_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y = x + 1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 2<\/h3>\n<p> Encontre todas as ass\u00edntotas obl\u00edquas da seguinte fun\u00e7\u00e3o racional: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-144807b8c72afbd43bb3f97d69cedb35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{2x^2-5}{x+3}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"118\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro, usamos a f\u00f3rmula para a inclina\u00e7\u00e3o da ass\u00edntota obl\u00edqua:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bc900ded359235b2293ec151e715daea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to \\pm\\infty} \\frac{f(x)}{x} = \\lim_{x \\to \\pm\\infty} \\cfrac{\\cfrac{2x^2-5}{x+3}}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"261\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Simplificamos a fra\u00e7\u00e3o aplicando as propriedades das fra\u00e7\u00f5es: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1c5afa9b1ca5f1c73e6b8e64c8fb9420_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to \\pm\\infty}\\frac{2x^2-5}{(x+3)\\cdot x}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"168\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-03a4b53a445bded103e8de4404620693_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to \\pm\\infty}\\frac{2x^2-5}{x^2+3x}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"149\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E determinamos o limite:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-461c274fc210474eddaf061463e92aaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to \\pm\\infty}\\frac{2x^2-5}{x^2+3x}= \\frac{+\\infty}{+\\infty} = \\frac{2}{1} = \\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"278\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O limite fornece um n\u00famero real diferente de zero, portanto \u00e9 uma fun\u00e7\u00e3o racional com uma ass\u00edntota obl\u00edqua cuja inclina\u00e7\u00e3o \u00e9 2.<\/p>\n<p class=\"has-text-align-left\"> Agora vamos calcular a intercepta\u00e7\u00e3o aplicando a f\u00f3rmula correspondente:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4a04a1abaebfc5e1781dd7d98399888e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[f(x)-mx\\right] = \\lim_{x \\to \\pm\\infty} \\left[\\frac{2x^2-5}{x+3}-2x\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"364\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Tentamos calcular o limite:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-00f35703d153fe6911328d143588e1cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[\\frac{2x^2-5}{x+3}-2x\\right]= \\bm{+\\infty - \\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"298\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mas obtemos a diferen\u00e7a indeterminada dos infinitos. Portanto, reduzimos os termos a um denominador comum e ent\u00e3o operamos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4920e8b21b180c4f2740ce712d9f30d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[\\frac{2x^2-5}{x+3}-2x\\right]=\\\\[6ex]=\\displaystyle\\lim_{x \\to \\pm\\infty} \\left[\\frac{2x^2-5}{x+3}-\\frac{2x\\cdot (x+3)}{x+3} \\right] = \\\\[6ex]=\\displaystyle\\lim_{x \\to \\pm\\infty} \\left[ \\frac{2x^2-5}{x+3}-\\frac{2x^2+6x}{x+3}\\right]=\\\\[6ex]=\\displaystyle\\lim_{x \\to \\pm\\infty}\\frac{2x^2-5-(2x^2+6x)}{x+3}\\\\[6ex]\\displaystyle =\\lim_{x \\to \\pm\\infty}\\frac{2x^2-5-2x^2-6x}{x+3}=\\\\[6ex]=\\displaystyle \\lim_{x \\to \\pm\\infty} \\frac{-6x-5}{x+3}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"434\" width=\"277\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E finalmente, resolvemos o limite:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-00b75da44399a44a4e215fd4baccf214_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n =\\lim_{x \\to \\pm\\infty} \\frac{-6x-5}{x+3}= \\frac{\\infty}{\\infty}=\\frac{-6}{1} = \\bm{-6}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"292\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Em resumo, a ass\u00edntota obl\u00edqua da fun\u00e7\u00e3o fracion\u00e1ria \u00e9: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6fbe1cc5f3362ddbd80ed0b29c0bb4ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ac6ac25ec7b85209d4d7d855e3d0b501_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y=2x-6}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Neste artigo explicamos o que s\u00e3o as ass\u00edntotas obl\u00edquas de uma fun\u00e7\u00e3o. Voc\u00ea aprender\u00e1 quando uma fun\u00e7\u00e3o tem uma ass\u00edntota obl\u00edqua e como ela \u00e9 calculada. E, al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos de ass\u00edntotas obl\u00edquas e praticar com exerc\u00edcios resolvidos passo a passo. O que \u00e9 uma ass\u00edntota obl\u00edqua? A ass\u00edntota obl\u00edqua de uma &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/assintota-obliqua\/\"> <span class=\"screen-reader-text\">Ass\u00edntota obl\u00edqua<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","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":"","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":"","footnotes":""},"categories":[16],"tags":[],"class_list":["post-22","post","type-post","status-publish","format-standard","hentry","category-limites-de-funcao"],"yoast_head":"<!-- This site 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