{"id":25,"date":"2023-09-17T11:05:21","date_gmt":"2023-09-17T11:05:21","guid":{"rendered":"https:\/\/mathority.org\/pt\/funcao-racional\/"},"modified":"2023-09-17T11:05:21","modified_gmt":"2023-09-17T11:05:21","slug":"funcao-racional","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/funcao-racional\/","title":{"rendered":"Fun\u00e7\u00e3o racional"},"content":{"rendered":"<p>Aqui voc\u00ea descobrir\u00e1 o que s\u00e3o fun\u00e7\u00f5es racionais. Al\u00e9m disso, explicamos como calcular o dom\u00ednio e as ass\u00edntotas de uma fun\u00e7\u00e3o racional. E n\u00e3o s\u00f3 isso, mas voc\u00ea ver\u00e1 quais s\u00e3o todas as caracter\u00edsticas das fun\u00e7\u00f5es racionais. Finalmente, voc\u00ea pode praticar exerc\u00edcios passo a passo sobre fun\u00e7\u00f5es racionais. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-una-funcion-racional\"><\/span> O que \u00e9 uma fun\u00e7\u00e3o racional?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A defini\u00e7\u00e3o de uma fun\u00e7\u00e3o racional \u00e9 a seguinte:<\/p>\n<p> <strong>Uma fun\u00e7\u00e3o racional \u00e9 uma fun\u00e7\u00e3o formada pelo quociente de dois polin\u00f4mios<\/strong> , ou seja, uma fun\u00e7\u00e3o racional \u00e9 uma fra\u00e7\u00e3o que possui um polin\u00f4mio no numerador e no denominador.<\/p>\n<p> As fun\u00e7\u00f5es racionais s\u00e3o caracterizadas por singularidades nos pontos onde o denominador desaparece.<\/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-a80ec33ef964f9463287fb8c93605b34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{a_0+a_1x+a_2x^2+\\dots +a_nx^n}{b_0+b_1x+b_2x^2+\\dots +n_nx^n}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"285\" style=\"vertical-align: -15px;\"><\/p>\n<\/p>\n<p> Fun\u00e7\u00f5es racionais tamb\u00e9m s\u00e3o chamadas de fun\u00e7\u00f5es fracion\u00e1rias.<\/p>\n<p> Por outro lado, as fun\u00e7\u00f5es racionais n\u00e3o devem ser confundidas com fun\u00e7\u00f5es irracionais. Fun\u00e7\u00f5es irracionais (ou radicais) s\u00e3o aquelas constitu\u00eddas por ra\u00edzes.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplos-de-funciones-racionales\"><\/span> Exemplos de fun\u00e7\u00f5es racionais<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para entender melhor a no\u00e7\u00e3o de fun\u00e7\u00e3o racional, veremos v\u00e1rios exemplos desse tipo de fun\u00e7\u00e3o.<\/p>\n<ul>\n<li> <u style=\"text-decoration-color:#FF9B28;\">Fun\u00e7\u00e3o racional com polin\u00f4mio de primeiro grau no numerador e denominador:<\/u><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b170c6c49759eeb2d1d3f81bbf0ebfc3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x+3}{2x-5}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"110\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Esses tipos de fun\u00e7\u00f5es racionais tamb\u00e9m s\u00e3o chamadas de <strong>fun\u00e7\u00f5es homogr\u00e1ficas<\/strong> .<\/p>\n<ul>\n<li> <u style=\"text-decoration-color:#FF9B28;\">Fun\u00e7\u00e3o racional com constante no numerador e polin\u00f4mio no denominador:<\/u><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-931f934a646a46832b66a8a3efe3ad17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{7}{x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"101\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Esses tipos de fun\u00e7\u00f5es racionais s\u00e3o chamados de <strong>fun\u00e7\u00f5es de proporcionalidade inversa<\/strong> e s\u00e3o usados para definir matematicamente quantidades inversamente proporcionais.<\/p>\n<ul>\n<li> <u style=\"text-decoration-color:#FF9B28;\">Fun\u00e7\u00e3o racional com um polin\u00f4mio de terceiro grau no numerador e um polin\u00f4mio de segundo grau no denominador:<\/u> <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-82bb59d3904629b47192dfd05456a638_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^3+4x^2-2x+6}{x^2+3x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"198\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"dominio-de-una-funcion-racional\"><\/span> Dom\u00ednio de uma fun\u00e7\u00e3o racional <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p> Um n\u00famero dividido por 0 \u00e9 uma indetermina\u00e7\u00e3o que d\u00e1 infinito (\u221e), ent\u00e3o uma fun\u00e7\u00e3o racional sempre existir\u00e1, a menos que o denominador seja 0.<\/p>\n<p> Portanto, <strong>o dom\u00ednio de uma fun\u00e7\u00e3o racional<\/strong> consiste em todos os n\u00fameros reais, exceto valores que cancelam o denominador.<\/p>\n<\/div>\n<p> Ent\u00e3o, para obter o dom\u00ednio de uma fun\u00e7\u00e3o racional, precisamos descobrir quando o denominador \u00e9 0, pois este ponto ser\u00e1 o \u00fanico que n\u00e3o pertence ao dom\u00ednio<\/p>\n<p> Vamos ver como o dom\u00ednio de uma fun\u00e7\u00e3o racional \u00e9 calculado resolvendo um exemplo:<\/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-dc119bb22722de1d946894030ac0e6e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{5x}{x+2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"101\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Primeiro definimos o denominador igual a 0 e depois resolvemos a equa\u00e7\u00e3o resultante:<\/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-24e301fa7ea2e8d9f0041192d9a84927_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x+2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"73\" style=\"vertical-align: -2px;\"><\/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-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, quando x for -2, o denominador ser\u00e1 0 e, portanto, a fun\u00e7\u00e3o n\u00e3o existir\u00e1. O dom\u00ednio da fun\u00e7\u00e3o, portanto, consiste em todos os n\u00fameros reais, exceto x=-2. Isto \u00e9 afirmado da seguinte 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-c69046d9ec5ea032ce1e6f7f070dbf83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f= \\mathbb{R}-\\{ -2 \\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"asintotas-de-una-funcion-racional\"><\/span> Ass\u00edntotas de uma fun\u00e7\u00e3o racional<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Uma das principais propriedades das fun\u00e7\u00f5es racionais s\u00e3o as suas ass\u00edntotas, pois determinam a sua representa\u00e7\u00e3o gr\u00e1fica.<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/representacao-de-funcoes\/\">representa\u00e7\u00e3o gr\u00e1fica de uma fun\u00e7\u00e3o<\/a><\/span><\/p>\n<p> As <strong>ass\u00edntotas de uma fun\u00e7\u00e3o racional<\/strong> s\u00e3o retas que o gr\u00e1fico da fun\u00e7\u00e3o se aproxima indefinidamente, mas nunca toca.<\/p>\n<p> Existem tr\u00eas tipos de ass\u00edntotas: ass\u00edntotas verticais, ass\u00edntotas horizontais e ass\u00edntotas obl\u00edquas.<\/p>\n<p> Abaixo voc\u00ea tem os tr\u00eas tipos de ass\u00edntotas que uma fun\u00e7\u00e3o racional pode representar graficamente em vermelho.<\/p>\n<p class=\"has-text-align-center\"> <u style=\"text-decoration-color:#E74C3C;\"><strong>Ass\u00edntota vertical de uma fun\u00e7\u00e3o racional<\/strong><\/u> <\/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\/exemple-dasymptote-verticale.webp\" alt=\"ass\u00edntota vertical de uma fun\u00e7\u00e3o racional\" class=\"wp-image-1294\" width=\"408\" height=\"361\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-center\"> <u style=\"text-decoration-color:#E74C3C;\"><strong>Ass\u00edntota horizontal de uma fun\u00e7\u00e3o racional<\/strong><\/u> <\/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\/exemple-dasymptote-horizontale.webp\" alt=\"ass\u00edntota horizontal de uma fun\u00e7\u00e3o racional\" class=\"wp-image-1333\" width=\"478\" height=\"378\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-center\"> <u style=\"text-decoration-color:#E74C3C;\"><strong>Ass\u00edntota obl\u00edqua de uma fun\u00e7\u00e3o racional<\/strong><\/u> <\/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=\"ass\u00edntota obl\u00edqua de uma fun\u00e7\u00e3o racional\" class=\"wp-image-1374\" width=\"386\" height=\"436\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Como voc\u00ea pode ver, determinar a ass\u00edntota de uma fun\u00e7\u00e3o a partir de seu gr\u00e1fico \u00e9 bastante simples, mas calcular as ass\u00edntotas de uma fun\u00e7\u00e3o racional sem ter sua representa\u00e7\u00e3o gr\u00e1fica \u00e9 bastante complicado. \u00c9 por isso que recomendamos que voc\u00ea veja como as ass\u00edntotas de uma fun\u00e7\u00e3o s\u00e3o calculadas em nosso site. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"caracteristicas-de-una-funcion-racional\"><\/span> Caracter\u00edsticas de uma fun\u00e7\u00e3o racional<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> As fun\u00e7\u00f5es racionais possuem as seguintes caracter\u00edsticas:<\/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-9c24e4e9a6871d3e8e07e85c24b039c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{P(x)}{Q(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"98\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li> <span style=\"color:#101010;font-weight: normal;\">Como vimos acima, o dom\u00ednio das fun\u00e7\u00f5es racionais inclui todos os n\u00fameros reais exceto valores que anulam o denominador da fra\u00e7\u00e3o.<\/span><\/li>\n<\/ul>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li> <span style=\"color:#101010;font-weight: normal;\">Em geral, o contradom\u00ednio (ou contradom\u00ednio) de uma fun\u00e7\u00e3o racional inclui todos os n\u00fameros reais, exceto valores em que a fun\u00e7\u00e3o possui uma ass\u00edntota horizontal.<\/span><\/li>\n<\/ul>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li> <span style=\"color:#101010;font-weight: normal;\">As fun\u00e7\u00f5es racionais s\u00e3o cont\u00ednuas em todo o seu dom\u00ednio. Ou seja, as fun\u00e7\u00f5es racionais apresentam descontinuidades em pontos que n\u00e3o pertencem ao seu dom\u00ednio.<\/span><\/li>\n<\/ul>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li> <span style=\"color:#101010;font-weight: normal;\">A representa\u00e7\u00e3o gr\u00e1fica da maioria das fun\u00e7\u00f5es racionais consiste em duas hip\u00e9rboles.<\/span><\/li>\n<\/ul>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li> <span style=\"color:#101010;font-weight: normal;\">Algumas regras para as ass\u00edntotas de fun\u00e7\u00f5es racionais podem ser deduzidas do numerador polinomial.\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> e o polin\u00f4mio denominador <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4d38083076c97f0893079e8fed89adb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"><\/p>\n<ul style=\"list-style-type:circle;margin-left:10%;color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:15px; margin-top:15px\"> <span style=\"color:#101010;font-weight: normal;\">Uma fun\u00e7\u00e3o racional tem uma ass\u00edntota vertical nos pontos que s\u00e3o as ra\u00edzes de\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8061e215a5d055a2cf14c44c4febfad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> mas estas n\u00e3o s\u00e3o ra\u00edzes de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0f6500c41747705211eacbfc8d05aba4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li style=\"margin-bottom:15px; margin-top:15px\"> <span style=\"color:#101010;font-weight: normal;\">Se o grau de\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> \u00e9 menor que o grau de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8061e215a5d055a2cf14c44c4febfad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> , a reta y=0 \u00e9 uma ass\u00edntota horizontal da fun\u00e7\u00e3o racional.<\/li>\n<li style=\"margin-bottom:15px; margin-top:15px\"> <span style=\"color:#101010;font-weight: normal;\">Se o grau de\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> \u00e9 maior que o grau de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8061e215a5d055a2cf14c44c4febfad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> , a fun\u00e7\u00e3o racional n\u00e3o tem ass\u00edntota horizontal.<\/li>\n<li> <span style=\"color:#101010;font-weight: normal;\">Se o grau de\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> \u00e9 uma unidade maior que o grau de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8061e215a5d055a2cf14c44c4febfad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> e os dois polin\u00f4mios n\u00e3o t\u00eam raiz comum, a fun\u00e7\u00e3o racional tem uma ass\u00edntota obl\u00edqua. <\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-funciones-racionales\"><\/span> Exerc\u00edcios resolvidos sobre fun\u00e7\u00f5es racionais<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Encontre o dom\u00ednio 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-94fcabf09d798d56e4d439c3dc4945b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\displaystyle\\frac{4x}{2x+4}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"110\" 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\"> \u00c9 uma fun\u00e7\u00e3o racional, ent\u00e3o o dom\u00ednio consiste em todos os n\u00fameros exceto aqueles que cancelam o denominador, porque ent\u00e3o a fun\u00e7\u00e3o daria \u221e.<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o definimos o denominador inteiro igual a zero para ver qual n\u00famero n\u00e3o pertence ao dom\u00ednio:<\/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-198bb8c11e20e5c9864ef9e60a2facc3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"82\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E resolvemos a equa\u00e7\u00e3o resultante: <\/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-8bb8c1283df065c83c44b7fe484324a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x=-4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"66\" style=\"vertical-align: 0px;\"><\/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-578104a63c70bc5ba4b685855966f28e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{-4}{2}=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"112\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O dom\u00ednio da fun\u00e7\u00e3o \u00e9, portanto, composto apenas por n\u00fameros, exceto -2: <\/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-ee75b8bb1136ab715a80e56e910f1626_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mathbf{Dom } \\ \\bm{f = \\mathbb{R}- \\{ -2 \\} }\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -5px;\"><\/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 os pontos de corte da seguinte fun\u00e7\u00e3o racional com os eixos cartesianos: <\/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-6ef48653e91a2935a9776b62ddd1f25b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^2-9}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"109\" style=\"vertical-align: -12px;\"><\/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\"> <strong>Ponto de corte com eixo X<\/strong><\/p>\n<p class=\"has-text-align-left\"> Para encontrar o ponto de intersec\u00e7\u00e3o da fun\u00e7\u00e3o com o eixo X \u00e9 necess\u00e1rio resolver <\/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-05bb421b504b7ae4aa483574cd6f28d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0:\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/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-0bce6c022ed0fc63f4659af75888f96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" style=\"vertical-align: -5px;\"><\/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-603043b6d5768eaace4011208f30bec1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2-9}{x}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"81\" 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-0b2fd4733c1dfbb47969d5b92e3e4f04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-9=0\\cdot x\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"104\" style=\"vertical-align: 0px;\"><\/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-ce55adbc277e9378607d68bce8ef19fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-9=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"81\" style=\"vertical-align: 0px;\"><\/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-05112cb5a98f653cd1920fb40e5ef9a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=9\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: 0px;\"><\/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-9cba0400a71268c96427f3b00bf29b6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=\\pm 3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"64\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Obtivemos duas solu\u00e7\u00f5es da equa\u00e7\u00e3o quadr\u00e1tica, portanto a fun\u00e7\u00e3o racional intercepta o eixo das abcissas em dois pontos diferentes, que s\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-013260528208aa1656c5407fa8e29db9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(3,0)\\qquad (-3,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Ponto de corte com eixo Y<\/strong><\/p>\n<p class=\"has-text-align-left\"> Para encontrar o ponto de intersec\u00e7\u00e3o com o eixo Y voc\u00ea deve calcular <\/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-8f92d7beea0ed3a053927c2d429d3450_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"><\/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-969a7e35c182b2950e797fec58ddab28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=\\cfrac{0^2-9}{0}=\\cfrac{-9}{0}= \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"203\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Qualquer n\u00famero dividido por zero \u00e9 uma indetermina\u00e7\u00e3o que d\u00e1 infinito. Portanto, a fun\u00e7\u00e3o racional n\u00e3o passa em nenhum ponto acima do eixo Y, ou seja, n\u00e3o possui ponto de intersec\u00e7\u00e3o com o eixo y.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 3<\/h3>\n<p> Trace a seguinte fun\u00e7\u00e3o racional em um gr\u00e1fico: <\/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-379664d569f63739a52aef2f4a3da41b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{2x+3}{2x+6}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"85\" 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\"> A primeira coisa a fazer \u00e9 calcular o dom\u00ednio da fun\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-eb310e295335d320e66cac6a8a6a3270_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x+6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"82\" style=\"vertical-align: -2px;\"><\/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-6b254eeeabf14c903b414b7f844bcd54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x =-6\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"66\" style=\"vertical-align: 0px;\"><\/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-be0dac801e36b79ec2bac9a5be70ad7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x =\\cfrac{-6}{2} =-3\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"113\" 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-3eb9671adf4127bd8129820378cb2a44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f = \\mathbb{R} - \\{ -3 \\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Depois de conhecermos o dom\u00ednio da fun\u00e7\u00e3o, constru\u00edmos uma tabela de valores:<\/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-5dfd65f4a7fca984bdc6f16ec89154c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c|c} x &amp; y \\\\ \\hline -2,5 &amp; -2 \\\\ -2 &amp; -0,5 \\\\ -1 &amp; 0,25 \\\\ 1 &amp; 0,63 \\\\ -3,5 &amp; 4  \\\\ -4 &amp; 2,5 \\\\ -5 &amp; 1,75 \\\\ -7 &amp; 1,38\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"200\" width=\"112\" style=\"vertical-align: -95px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para finalizar, basta representar os pontos obtidos em um gr\u00e1fico e desenhar as hip\u00e9rboles, desenhando assim a fun\u00e7\u00e3o racional: <\/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\/fonction-de-proportionnalite-inverse.webp\" alt=\"fun\u00e7\u00e3o de proporcionalidade inversa\" class=\"wp-image-170\" width=\"545\" height=\"460\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 4<\/h3>\n<p> Determine as ass\u00edntotas da fun\u00e7\u00e3o racional representada graficamente abaixo: <\/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\/fonction-representation-limites-infini.webp\" alt=\"ass\u00edntotas de uma fun\u00e7\u00e3o racional\" class=\"wp-image-1244\" width=\"403\" height=\"406\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\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 s\u00e3o claramente vis\u00edveis no gr\u00e1fico, pois s\u00e3o representadas como linhas pontilhadas vermelhas.<\/p>\n<p class=\"has-text-align-left\"> Neste problema, a fun\u00e7\u00e3o est\u00e1 muito pr\u00f3xima da linha horizontal y=1, mas nunca a toca. Portanto, a fun\u00e7\u00e3o racional possui uma \u00fanica ass\u00edntota horizontal, que \u00e9 y=1.<\/p>\n<p class=\"has-text-align-left\"> Da mesma forma, a representa\u00e7\u00e3o gr\u00e1fica da fun\u00e7\u00e3o est\u00e1 muito pr\u00f3xima das linhas verticais x=-1 e x=1, mas nunca atinge estes valores. A fun\u00e7\u00e3o racional, portanto, tem duas ass\u00edntotas verticais diferentes, que s\u00e3o x=-1 e x=1.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 5<\/h3>\n<p> Calcule todas as ass\u00edntotas 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-ffd06824234d445d38d021cbb04bfa23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{6x-4}{2x+2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"110\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> <strong>Nota:<\/strong> Para resolver este exerc\u00edcio, recomendamos que voc\u00ea primeiro acesse o link acima sobre <u style=\"text-decoration-color:#FF9B28;\">como s\u00e3o calculadas as ass\u00edntotas de uma fun\u00e7\u00e3o<\/u> e veja a explica\u00e7\u00e3o. <\/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\"> <strong><span style=\"text-decoration: underline;\">ass\u00edntota vertical<\/span><\/strong><\/p>\n<p class=\"has-text-align-left\"> Para calcular as ass\u00edntotas verticais de uma fun\u00e7\u00e3o, devemos primeiro encontrar o dom\u00ednio da fun\u00e7\u00e3o. Portanto, igualamos o denominador da fun\u00e7\u00e3o racional a 0 para encontrar os pontos que n\u00e3o pertencem ao dom\u00ednio: <\/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-ba1a17401e951a8539e475d758a871d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x +2 =0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"82\" style=\"vertical-align: -2px;\"><\/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-b2a522ee7d1c1819c496c45af9549bc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x= -2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"65\" style=\"vertical-align: 0px;\"><\/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-3e60b04854152fc93f76ad6c29e09346_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = \\cfrac{-2}{2} = -1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"112\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O dom\u00ednio da fun\u00e7\u00e3o, portanto, consiste em todos os n\u00fameros, exceto -1:<\/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-09f86d513e25805efa3dbddfc2e0229e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f = \\mathbb{R} - \\left\\{ -1 \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o x=-1 poderia ser uma ass\u00edntota vertical. Para verificar isso, devemos calcular o limite da fun\u00e7\u00e3o no ponto:<\/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-19ee5b45a22b402ee890392a91803649_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -1 } \\frac{6x-4}{2x+2} = \\frac{6\\cdot(-1)-4}{2\\cdot(-1)+2}=\\frac{-10}{0}= \\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"311\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, x=-1 \u00e9 uma ass\u00edntota vertical da fun\u00e7\u00e3o racional, j\u00e1 que o limite da fun\u00e7\u00e3o neste ponto d\u00e1 infinito.<\/p>\n<p class=\"has-text-align-left\"> <strong><span style=\"text-decoration: underline;\">ass\u00edntota horizontal<\/span><\/strong><\/p>\n<p class=\"has-text-align-left\"> Para determinar as ass\u00edntotas horizontais, precisamos calcular o limite infinito da fun\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-604ce9e6e3a0943003f79d5f890b81d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\frac{6x-4}{2x+2} = \\frac{6(+\\infty)}{2(+\\infty)} = \\frac{+\\infty}{+\\infty} = \\frac{6}{2} = \\bm{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"312\" 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-823441caef29e22ea5cda9685af7c1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -\\infty}\\frac{6x-4}{2x+2} = \\frac{6(-\\infty)}{2(-\\infty)} = \\frac{-\\infty}{-\\infty} = \\frac{6}{2} = \\bm{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"312\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nesse caso, o resultado do limite infinito indeterminado entre o infinito \u00e9 a divis\u00e3o dos coeficientes do x de maior grau, pois o numerador e o denominador s\u00e3o da mesma ordem.<\/p>\n<p class=\"has-text-align-left\"> Os dois limites infinitos da fun\u00e7\u00e3o nos deram 3, ent\u00e3o y=3 \u00e9 uma ass\u00edntota horizontal da fun\u00e7\u00e3o racional.<\/p>\n<p class=\"has-text-align-left\"> <strong><span style=\"text-decoration: underline;\">ass\u00edntota obl\u00edqua<\/span><\/strong><\/p>\n<p class=\"has-text-align-left\"> Como existe uma ass\u00edntota horizontal, a fun\u00e7\u00e3o racional n\u00e3o possui uma ass\u00edntota obl\u00edqua.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Aqui voc\u00ea descobrir\u00e1 o que s\u00e3o fun\u00e7\u00f5es racionais. Al\u00e9m disso, explicamos como calcular o dom\u00ednio e as ass\u00edntotas de uma fun\u00e7\u00e3o racional. E n\u00e3o s\u00f3 isso, mas voc\u00ea ver\u00e1 quais s\u00e3o todas as caracter\u00edsticas das fun\u00e7\u00f5es racionais. Finalmente, voc\u00ea pode praticar exerc\u00edcios passo a passo sobre fun\u00e7\u00f5es racionais. O que \u00e9 uma fun\u00e7\u00e3o racional? A &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/funcao-racional\/\"> <span class=\"screen-reader-text\">Fun\u00e7\u00e3o racional<\/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":[22],"tags":[],"class_list":["post-25","post","type-post","status-publish","format-standard","hentry","category-representacao-de-funcao"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Fun\u00e7\u00e3o racional: o que \u00e9, dom\u00ednio, ass\u00edntotas, gr\u00e1fico, exerc\u00edcios,...<\/title>\n<meta name=\"description\" content=\"Explicamos o que \u00e9 uma fun\u00e7\u00e3o racional e todas as suas caracter\u00edsticas (dom\u00ednio, ass\u00edntotas, gr\u00e1fico, etc.). Com exerc\u00edcios resolvidos sobre fun\u00e7\u00f5es racionais.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/funcao-racional\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Fun\u00e7\u00e3o racional: o que \u00e9, dom\u00ednio, ass\u00edntotas, gr\u00e1fico, exerc\u00edcios,...\" \/>\n<meta property=\"og:description\" content=\"Explicamos o que \u00e9 uma fun\u00e7\u00e3o racional e todas as suas caracter\u00edsticas (dom\u00ednio, ass\u00edntotas, gr\u00e1fico, etc.). Com exerc\u00edcios resolvidos sobre fun\u00e7\u00f5es racionais.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/funcao-racional\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-17T11:05:21+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80ec33ef964f9463287fb8c93605b34_l3.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"7 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/funcao-racional\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/funcao-racional\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Fun\u00e7\u00e3o racional\",\"datePublished\":\"2023-09-17T11:05:21+00:00\",\"dateModified\":\"2023-09-17T11:05:21+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/funcao-racional\/\"},\"wordCount\":1488,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Representa\u00e7\u00e3o de fun\u00e7\u00e3o\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/funcao-racional\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/funcao-racional\/\",\"url\":\"https:\/\/mathority.org\/pt\/funcao-racional\/\",\"name\":\"Fun\u00e7\u00e3o racional: o que \u00e9, dom\u00ednio, ass\u00edntotas, gr\u00e1fico, exerc\u00edcios,...\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/#website\"},\"datePublished\":\"2023-09-17T11:05:21+00:00\",\"dateModified\":\"2023-09-17T11:05:21+00:00\",\"description\":\"Explicamos o que \u00e9 uma fun\u00e7\u00e3o racional e todas as suas caracter\u00edsticas (dom\u00ednio, ass\u00edntotas, gr\u00e1fico, etc.). Com exerc\u00edcios resolvidos sobre fun\u00e7\u00f5es racionais.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/pt\/funcao-racional\/#breadcrumb\"},\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/pt\/funcao-racional\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/pt\/funcao-racional\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Fun\u00e7\u00e3o racional\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/pt\/#website\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"name\":\"Mathority\",\"description\":\"Onde a curiosidade encontra o c\u00e1lculo!\",\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/pt\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pt-BR\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathority.org\/pt\/#organization\",\"name\":\"Mathority\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"contentUrl\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"width\":703,\"height\":151,\"caption\":\"Mathority\"},\"image\":{\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\",\"name\":\"Equipe Mathoridade\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Equipe Mathoridade\"},\"sameAs\":[\"http:\/\/mathority.org\/pt\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Fun\u00e7\u00e3o racional: o que \u00e9, dom\u00ednio, ass\u00edntotas, gr\u00e1fico, exerc\u00edcios,...","description":"Explicamos o que \u00e9 uma fun\u00e7\u00e3o racional e todas as suas caracter\u00edsticas (dom\u00ednio, ass\u00edntotas, gr\u00e1fico, etc.). Com exerc\u00edcios resolvidos sobre fun\u00e7\u00f5es racionais.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/funcao-racional\/","og_locale":"pt_BR","og_type":"article","og_title":"Fun\u00e7\u00e3o racional: o que \u00e9, dom\u00ednio, ass\u00edntotas, gr\u00e1fico, exerc\u00edcios,...","og_description":"Explicamos o que \u00e9 uma fun\u00e7\u00e3o racional e todas as suas caracter\u00edsticas (dom\u00ednio, ass\u00edntotas, gr\u00e1fico, etc.). Com exerc\u00edcios resolvidos sobre fun\u00e7\u00f5es racionais.","og_url":"https:\/\/mathority.org\/pt\/funcao-racional\/","article_published_time":"2023-09-17T11:05:21+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80ec33ef964f9463287fb8c93605b34_l3.png"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"7 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/funcao-racional\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/funcao-racional\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Fun\u00e7\u00e3o racional","datePublished":"2023-09-17T11:05:21+00:00","dateModified":"2023-09-17T11:05:21+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/funcao-racional\/"},"wordCount":1488,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Representa\u00e7\u00e3o de fun\u00e7\u00e3o"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/funcao-racional\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/funcao-racional\/","url":"https:\/\/mathority.org\/pt\/funcao-racional\/","name":"Fun\u00e7\u00e3o racional: o que \u00e9, dom\u00ednio, ass\u00edntotas, gr\u00e1fico, exerc\u00edcios,...","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/#website"},"datePublished":"2023-09-17T11:05:21+00:00","dateModified":"2023-09-17T11:05:21+00:00","description":"Explicamos o que \u00e9 uma fun\u00e7\u00e3o racional e todas as suas caracter\u00edsticas (dom\u00ednio, ass\u00edntotas, gr\u00e1fico, etc.). Com exerc\u00edcios resolvidos sobre fun\u00e7\u00f5es racionais.","breadcrumb":{"@id":"https:\/\/mathority.org\/pt\/funcao-racional\/#breadcrumb"},"inLanguage":"pt-BR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/pt\/funcao-racional\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/pt\/funcao-racional\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/pt\/"},{"@type":"ListItem","position":2,"name":"Fun\u00e7\u00e3o racional"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/pt\/#website","url":"https:\/\/mathority.org\/pt\/","name":"Mathority","description":"Onde a curiosidade encontra o c\u00e1lculo!","publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/pt\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"pt-BR"},{"@type":"Organization","@id":"https:\/\/mathority.org\/pt\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/pt\/","logo":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00","name":"Equipe Mathoridade","image":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Equipe Mathoridade"},"sameAs":["http:\/\/mathority.org\/pt"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/25","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/comments?post=25"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/25\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/media?parent=25"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/categories?post=25"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/tags?post=25"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}