{"id":374,"date":"2023-07-04T04:08:40","date_gmt":"2023-07-04T04:08:40","guid":{"rendered":"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/"},"modified":"2023-07-04T04:08:40","modified_gmt":"2023-07-04T04:08:40","slug":"indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/","title":{"rendered":"Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e)"},"content":{"rendered":"<p>Neste artigo explicamos como resolver a indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e). Voc\u00ea encontrar\u00e1 exemplos dessa indetermina\u00e7\u00e3o com diversos tipos de fun\u00e7\u00f5es e, al\u00e9m disso, poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo de indetermina\u00e7\u00e3o infinita menos infinita. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"resolver-la-indeterminacion-infinito-menos-infinito\"><\/span> Resolvendo indetermina\u00e7\u00e3o infinita menos infinita<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Quando o limite de uma fun\u00e7\u00e3o d\u00e1 infinito menos infinito, significa que \u00e9 uma indetermina\u00e7\u00e3o (ou uma forma indeterminada). Ou seja <strong>, o limite de uma fun\u00e7\u00e3o que d\u00e1 indetermina\u00e7\u00e3o menos infinito<\/strong> n\u00e3o pode ser determinado realizando o c\u00e1lculo direto, mas sim um procedimento preliminar deve ser realizado.<\/p>\n<p> Portanto, para <strong>resolver a indetermina\u00e7\u00e3o infinita menos infinita,<\/strong> devemos primeiro aplicar um procedimento que depende do tipo de fun\u00e7\u00e3o: se for uma fun\u00e7\u00e3o polinomial, pode ser calculada por compara\u00e7\u00e3o, se for uma fun\u00e7\u00e3o racional, as fra\u00e7\u00f5es devem ser reduzidas a um denominador comum e, se for uma fun\u00e7\u00e3o irracional, deve ser multiplicado pelo conjugado.<\/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-b1bfc56d2079c86e8ad6e1943311b730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to+\\infty}\\Bigl(f(x)-g(x)\\Bigr)=\\infty-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"236\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> A seguir veremos com exemplos como a indetermina\u00e7\u00e3o infinito menos infinito \u00e9 resolvida em cada tipo de fun\u00e7\u00e3o. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-infinito-menos-infinito-en-funciones-polinomicas\"><\/span> Indetermina\u00e7\u00e3o infinita menos infinita em fun\u00e7\u00f5es polinomiais <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 style=\"text-align:left\"> Em um polin\u00f4mio, a indetermina\u00e7\u00e3o infinito menos infinito \u00e9 igual ao infinito de maior ordem, ou seja, o termo de maior ordem determina o sinal positivo ou negativo do infinito.<\/p>\n<\/div>\n<p> Por exemplo, observe o limite da seguinte fun\u00e7\u00e3o polinomial que fornece a forma indeterminada infinito menos infinito:<\/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-f61115731d29dc9f05941968417c9443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to+\\infty}\\bigl(x^2-3x\\bigr)=(+\\infty)^2-3\\cdot (\\infty)=+\\infty-\\infty=+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"421\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Neste caso, o termo x <sup>2<\/sup> \u00e9 de segundo grau e o termo 3x \u00e9 de primeiro grau, portanto o mon\u00f4mio x <sup>2<\/sup> \u00e9 dominante por ser de ordem superior. Portanto, o resultado do limite \u00e9 o infinito obtido deste termo.<\/p>\n<p> D\u00ea uma olhada nestes outros exemplos:<\/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-3bfb5cd294a19de382f74738af6be724_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to+\\infty}\\bigl(x^5-4x^2-3x\\bigr)=(+\\infty)^5=+\\infty\\\\[5ex]\\displaystyle\\lim_{x\\to-\\infty}\\bigl(-3x^2-5x\\bigr)=-3\\cdot (-\\infty)^2=-3\\cdot \\infty=-\\infty\\\\[5ex]\\displaystyle\\lim_{x\\to+\\infty}\\bigl(x^7-5x^4+x^3-2x-10\\bigr)=(+\\infty)^7=+\\infty\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"150\" width=\"387\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Em suma, quando estabelecemos limites ao infinito em fun\u00e7\u00f5es polinomiais <strong>, devemos simplesmente substituir o infinito no termo de maior grau<\/strong> , ignorando todos os outros termos. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-infinito-menos-infinito-con-fracciones\"><\/span> Indetermina\u00e7\u00e3o infinita menos infinita com fra\u00e7\u00f5es <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 style=\"text-align:left\"> Quando <strong>ocorre a indetermina\u00e7\u00e3o infinito menos infinito em uma adi\u00e7\u00e3o ou subtra\u00e7\u00e3o de fra\u00e7\u00f5es alg\u00e9bricas<\/strong> , devemos primeiro fazer a adi\u00e7\u00e3o ou subtra\u00e7\u00e3o das fra\u00e7\u00f5es e depois calcular o limite.<\/p>\n<\/div>\n<p> Vamos ver como calcular a indetermina\u00e7\u00e3o infinito menos infinito em uma fun\u00e7\u00e3o com fra\u00e7\u00f5es resolvendo um exemplo passo a passo:<\/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-c58eb86af2eb0393a802fc7a29f8a453_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\left( \\frac{x^2}{x-1} - \\frac{x}{3}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"152\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Vamos tentar calcular o limite primeiro:<\/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-b3a2cbbfec28f9de05668b90e9ee65f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\left(  \\frac{x^2}{x-1} - \\frac{x}{3}\\right) = \\frac{(+\\infty)^2}{(+\\infty)-1} - \\frac{+\\infty}{3} = \\bm{+\\infty - \\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"410\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Mas obtemos a indetermina\u00e7\u00e3o \u221e-\u221e.<\/p>\n<p> Ent\u00e3o primeiro precisamos fazer a subtra\u00e7\u00e3o de fra\u00e7\u00f5es. Para isso, reduzimos as fra\u00e7\u00f5es a um denominador comum, ou seja, multiplicamos o numerador e o denominador de uma fra\u00e7\u00e3o pelo denominador da outra:<\/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-68e489c5833478cb20929ea07ae2971d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x \\to +\\infty} \\left( \\frac{x^2}{x-1}-\\frac{x}{3}\\right)=\\\\[5ex]\\displaystyle =\\lim_{x \\to +\\infty}\\left(\\frac{x^2 \\cdot 3}{(x-1)\\cdot 3}- \\frac{x\\cdot (x-1)}{3\\cdot (x-1)} \\right)=\\\\[5ex]\\displaystyle = \\lim_{x \\to +\\infty} \\left( \\frac{3x^2 }{3(x-1)}- \\frac{x^2-x}{3(x-1)}\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"188\" width=\"295\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E agora que ambas as fra\u00e7\u00f5es t\u00eam o mesmo denominador, podemos combin\u00e1-las numa \u00fanica 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-1e5345a6d68ae0cdda543b81f89daa62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\frac{3x^2 -(x^2-x)}{3(x-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"163\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Operamos no numerador e no denominador:<\/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-31cbae0091a641d74250fae5758b3116_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}  \\frac{3x^2 -x^2+x}{3x-3} =  \\lim_{x \\to +\\infty}  \\frac{2x^2+x}{3x-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"284\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> E finalmente, calculamos o limite novamente:<\/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-6ef29c026035a5353b2bada5bc0d9ff9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\frac{2x^2+x}{3x-3}=\\frac{+\\infty}{+\\infty}=\\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"225\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Neste caso a indetermina\u00e7\u00e3o infinita entre o infinito d\u00e1 +\u221e porque o grau do numerador \u00e9 maior que o grau do denominador.<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <a href=\"https:\/\/mathority.org\/pt\/indeterminacao-infinita-entre-infinito-%e2%88%9e-%e2%88%9e\/\"><span style=\"text-decoration: underline;\">o que \u00e9 o infinito entre o infinito?<\/span><\/a> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-infinito-menos-infinito-con-raices\"><\/span> indetermina\u00e7\u00e3o infinito menos infinito com ra\u00edzes <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 style=\"text-align:left\"> Quando <strong>a indetermina\u00e7\u00e3o infinito menos infinito ocorre na adi\u00e7\u00e3o ou subtra\u00e7\u00e3o radical<\/strong> , devemos primeiro multiplicar e dividir a fun\u00e7\u00e3o pela express\u00e3o radical conjugada e, em seguida, resolver o limite.<\/p>\n<\/div>\n<p> Veremos como resolver a indetermina\u00e7\u00e3o infinito menos infinito em uma fun\u00e7\u00e3o irracional usando um exemplo passo a passo:<\/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-e093b62c357684fe8a8818df58d7b99a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(x-\\sqrt{x^2-5}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"165\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Vamos primeiro tentar resolver o limite da fun\u00e7\u00e3o com radicais:<\/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-4459c2b6c968344878499cfbb30adda4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(x-\\sqrt{x^2-5}\\right)=+\\infty-\\sqrt{(+\\infty)^2}=\\bm{+\\infty-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"409\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> No entanto, obtemos a forma indeterminada \u221e-\u221e. Ent\u00e3o, para saber quanto de indetermina\u00e7\u00e3o \u00e9 infinito menos infinito, precisamos aplicar o procedimento explicado.<\/p>\n<p> Como a fun\u00e7\u00e3o possui radicais, multiplicamos e dividimos toda a fun\u00e7\u00e3o pela express\u00e3o irracional conjugada:<\/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-f10d91882a0f8dcca86fbb8dda7da7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\left(x-\\sqrt{x^2-5}\\right)= \\lim_{x \\to +\\infty}\\frac{\\left(x-\\sqrt{x^2-5}\\right)\\cdot\\left(x+\\sqrt{x^2-5}\\right)}{x+\\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"488\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> A express\u00e3o alg\u00e9brica do numerador corresponde \u00e0 identidade not\u00e1vel do produto de uma soma por uma diferen\u00e7a, podemos portanto simplificar a express\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-b00f177bdb579dabf9dc589e387344cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{\\left(x-\\sqrt{x^2-5}\\right) \\cdot \\left(x + \\sqrt{x^2-5}\\right)}{ x + \\sqrt{x^2-5}}= \\lim_{x \\to +\\infty} \\cfrac{x^2- \\left( \\sqrt{x^2-5}\\right)^2}{ x + \\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"505\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Agora simplificamos a raiz do limite, j\u00e1 que \u00e9 elevado ao quadrado:<\/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-d5c798f099ef1c56a50526e7fba8c99c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\frac{x^2-(x^2-5)}{x+\\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> Operamos no numerador 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-392ae211b16ad803eb70cc4993a0c7e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\frac{x^2- x^2+5}{x+\\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"146\" style=\"vertical-align: -16px;\"><\/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-be954eaf609b9f98c6dc984758599b5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\frac{5}{x+\\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"146\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> E por fim, refazemos o c\u00e1lculo do 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-c29edfa5eba2fe54e369c3d963d11a45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\frac{5}{x+\\sqrt{x^2-5}}=\\frac{5}{+\\infty+\\sqrt{(+\\infty)^2}}=\\frac{5}{+\\infty}=\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"391\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p> O resultado do limite \u00e9 portanto 0, porque qualquer n\u00famero dividido pelo infinito \u00e9 igual a zero. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-indeterminacion-infinito-menos-infinito\"><\/span> Resolvidos problemas de indetermina\u00e7\u00e3o infinita menos infinita<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Resolva o seguinte limite quando x se aproxima de mais infinito: <\/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-bac36c53d4e34c6e9972009b34a64c21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}(7x^2-2x^3)\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"134\" style=\"vertical-align: -13px;\"><\/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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste limite, o termo de ordem mais elevada \u00e9 de terceiro grau, por isso focamos no infinito obtido deste termo. <\/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-eee2b50312f5fd4225c85387c311eec5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}(+7x^2-2x^3)=+\\infty^2-\\infty^3=+\\infty-\\infty=\\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"408\" style=\"vertical-align: -13px;\"><\/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> Calcule o limite da seguinte fun\u00e7\u00e3o polinomial quando x se aproxima do infinito negativo: <\/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-12b903d03d726c28b625fe3f5ba4b3c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -\\infty}(-5x^3-9x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"148\" 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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> O infinito negativo ao cubo permanece negativo, mas quando elevado ao quadrado torna-se positivo. mais tarde Embora seus sinais sejam modificados pelos coeficientes \u00e0 sua frente:<\/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-172f2927f65e61079b13abd02234f1c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to -\\infty}(-5x^3+x^2)=\\\\[3ex]=-5(-\\infty)^3-9(-\\infty)^2=\\\\[3ex]=-5\\cdot (-\\infty)-9\\cdot \\infty=\\\\[3ex]=+\\infty-\\infty\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"153\" width=\"196\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o, a forma indeterminada infinito menos infinito \u00e9 definida pelo termo de ordem mais alta (-5x <sup>3<\/sup> ), do qual obtemos infinito positivo: <\/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-dfd806b31a588234442f48fa5ae8b751_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -\\infty}(-5x^3+x^2)=\\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"194\" style=\"vertical-align: -12px;\"><\/p>\n<\/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> Determine o limite ao infinito 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-f0a8401379d90875626b1fbd3714fd01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(\\frac{x^3+1}{x-1}-\\frac{x}{4} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"160\" style=\"vertical-align: -17px;\"><\/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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro, tentamos calcular o limite substituindo o infinito na 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-43df032007d76e00f2f7366e05f9e697_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(\\frac{x^3+1}{x-1}-\\frac{x}{4}\\right)=\\frac{(+\\infty)^3+1}{+\\infty-1}-\\frac{+\\infty}{4} = \\bm{+\\infty -\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"425\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mas acabamos com a indetermina\u00e7\u00e3o \u221e \u2013 \u221e. Portanto, reduzimos as fra\u00e7\u00f5es a um denominador comum:<\/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-7e2820674bc86d085f6deec7fdf9adf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim\\limits_{x \\to +\\infty} \\left(\\frac{x^3+1}{x-1}-\\frac{x}{4} \\right)=\\\\[5ex]\\displaystyle = \\lim_{x\\to +\\infty}\\left(\\frac{(x^3+1)\\cdot4}{(x-1)\\cdot4}-\\frac{x\\cdot(x-1)}{4\\cdot (x-1)}\\right)=\\\\[5ex]\\displaystyle =\\lim_{x\\to +\\infty}\\left(\\frac{4x^3+4}{4x-4}-\\frac{x^2-x}{4x-4}\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"188\" width=\"302\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E como ambas as fra\u00e7\u00f5es agora t\u00eam o mesmo denominador, podemos combin\u00e1-las em uma 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-93a00027be74b1e60c7ee8537ebe5d9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\left(\\frac{4x^3+4}{4x-4}-\\frac{x^2-x}{4x-4}\\right)=\\lim_{x\\to +\\infty}\\frac{4x^3+4-(x^2-x)}{4x-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"429\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Fazemos os par\u00eanteses do 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-c7de3ead5b3a5f8bd2ae8d767da693b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty} \\frac{4x^3+4-x^2+x}{4x-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"180\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, finalmente, 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-7ffb73fbf26fd2b625e43872a9c10ef9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\frac{4x^3+4-x^2+x}{4x-4}=\\frac{4(+\\infty)^3}{4(+\\infty)}=\\frac{+\\infty}{+\\infty} = \\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"384\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Neste caso a indetermina\u00e7\u00e3o \u221e\/\u221e d\u00e1 +\u221e porque o grau do numerador \u00e9 maior que o grau do denominador.<\/p>\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> Resolva o limite da seguinte fun\u00e7\u00e3o fracion\u00e1ria quando x se aproxima de 0: <\/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-e783bc22baa422d4b537fae4628fb4a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5}{x^2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"165\" style=\"vertical-align: -17px;\"><\/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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro tentamos calcular o limite normalmente:<\/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-207bc08385f430f0f8c49ac34a10f811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5}{x^2}\\right)=\\frac{-3\\cdot0-2}{0^4}-\\frac{5}{0^2}=\\frac{-2}{0}-\\frac{5}{0}=\\bm{\\infty-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"477\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mas obtemos a forma indeterminada \u221e-\u221e. Portanto, precisamos reduzir as fra\u00e7\u00f5es da fun\u00e7\u00e3o a um denominador comum.<\/p>\n<p class=\"has-text-align-left\"> Neste caso, x <sup>4<\/sup> \u00e9 um m\u00faltiplo de x <sup>2<\/sup> , ent\u00e3o simplesmente multiplicando o numerador e o denominador da segunda fra\u00e7\u00e3o por x <sup>2<\/sup> obteremos que ambas as fra\u00e7\u00f5es t\u00eam o mesmo denominador:<\/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-876115dc1fb49e81373d70be5fdcfb5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x \\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5}{x^2}\\right)=\\\\[5ex]\\displaystyle =\\lim_{x \\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5\\cdot x^2}{x^2\\cdot x^2} \\right)=\\\\[5ex]\\displaystyle =\\lim_{x\\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5x^2}{x^4}\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"186\" width=\"235\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora podemos subtrair as duas 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-bf56e81e075d9ac498e9df87a94a675f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5x^2}{x^4}\\right)=\\lim_{x\\to 0}\\frac{-3x-2-5x^2 }{x^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"346\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Tentamos resolver o limite novamente:<\/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-b231cf80ccb03d1287c1aab47769bc34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 0}  \\cfrac{-3x-2-5x^2 }{x^4} =\\cfrac{-3\\cdot 0-2-5\\cdot 0^2}{0^4}=\\frac{-2}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"370\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mas acabamos com a indetermina\u00e7\u00e3o de uma constante dividida por zero. \u00c9 portanto necess\u00e1rio calcular os limites laterais 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-c4ced459b1e0da92f03d9d9515b6ea68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 0^{-}} \\frac{-3x-2-5x^2}{x^4}=\\frac{-2}{+0}=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"262\" style=\"vertical-align: -14px;\"><\/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-239f065e0fe7bb4055e63a8477c030f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 0^{+}}\\frac{-3x-2-5x^2}{x^4}=\\frac{-2}{+0}=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"262\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Concluindo, como os dois limites laterais da fun\u00e7\u00e3o no ponto x=0 d\u00e3o -\u221e, a solu\u00e7\u00e3o do limite \u00e9 -\u221e: <\/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-30ab5fa39e1b25568d55de0cc4267dc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 0^-}f(x)=\\lim_{x \\to 0^+}f(x)=-\\infty\\ \\longrightarrow \\  \\lim_{x \\to 0}f(x)= \\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"401\" style=\"vertical-align: -14px;\"><\/p>\n<\/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> Resolva o limite ao infinito da seguinte fun\u00e7\u00e3o com ra\u00edzes: <\/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-5fb2be8c217ffddadf1b3d9d55f100c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(4x^2-\\sqrt{x^4+1}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"182\" style=\"vertical-align: -13px;\"><\/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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Tentando resolver o limite, obtemos a indetermina\u00e7\u00e3o infinito menos infinito:<\/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-8a1a6b3ff08a703378b8cfb1b5e6532c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(4x^2-\\sqrt{x^4+1}\\right)=4(+\\infty)^2-\\sqrt{(+\\infty)^4}=\\bm{+\\infty -\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"456\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, como existem radicais na fun\u00e7\u00e3o, precisamos multiplic\u00e1-la e dividi-la pela express\u00e3o radical conjugada:<\/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-3c4cdc9585a792800b8c903745ecc7c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\left(4x^2-\\sqrt{x^4+1} \\right)=\\lim_{x \\to +\\infty}\\frac{\\left(4x^2-\\sqrt{x^4+1}\\right)\\cdot\\left(4x^2+\\sqrt{x^4+1}\\right)}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"538\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> No numerador temos o produto not\u00e1vel de uma soma por uma diferen\u00e7a, que \u00e9 igual \u00e0 diferen\u00e7a dos quadrados. Ainda:<\/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-1aab32f1a28189a4ce96f3816f11a02e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\left(4x^2\\right)^2-\\left(\\sqrt{x^4+1}\\right)^2}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"216\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Simplificamos o radical ao quadrado:<\/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-6d86adc198c4fb2cd1d99c94e5b8430e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\bigl(4x^2\\bigr)^2-(x^4+1)}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"186\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> 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-5c07c403048b4d3e40a8034333ff069c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{16x^4-x^4-1}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"163\" style=\"vertical-align: -16px;\"><\/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-c138b064a8fa3142cb2d50782807ebb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{15x^4-1}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"163\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, finalmente, encontramos 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-cdb8845be6c640f0370961c3a52598d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{15x^4-1}{4x^2+\\sqrt{x^4+1}}=\\frac{15(+\\infty)^4}{4(+\\infty)^2+\\sqrt{(+\\infty)^4}}=\\frac{+\\infty}{+\\infty}= \\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"460\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Neste caso a indetermina\u00e7\u00e3o infinita dividida pelo infinito \u00e9 mais infinita porque o grau do numerador \u00e9 maior que o grau do denominador (lembre-se que a raiz quadrada reduz o grau em dois:<\/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-ded55d5413ed7bccc29e8228df205f19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{x^4} = x^{4\/2} = x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -1px;\"><\/p>\n<p> ).<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 6<\/h3>\n<p> Resolva o limite quando x se aproxima do infinito da seguinte fun\u00e7\u00e3o irracional: <\/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-7d9f21f0159778cdb1f0710e1a9e0023_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(2x-1-\\sqrt{4x^2+1}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"214\" style=\"vertical-align: -13px;\"><\/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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro, tentamos calcular o limite normalmente:<\/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-5419056e772f9d11884cae7e315ca947_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(2x-1-\\sqrt{4x^2+1}\\right)=2(+\\infty)-\\sqrt{4(+\\infty)^2}=\\bm{+\\infty -\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"489\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mas nos d\u00e1 como resultado a indetermina\u00e7\u00e3o da diferen\u00e7a dos infinitos. Portanto, como a fun\u00e7\u00e3o possui ra\u00edzes, precisamos multiplicar e dividir a express\u00e3o pelo radical conjugado:<\/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-bde8a1f86cf7be80170b9595b5a822df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\left(2x-1-\\sqrt{4x^2+1}\\right)\\cdot\\left(2x-1+\\sqrt{4x^2+1}\\right)}{2x-1 +\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"393\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agrupamos a igualdade not\u00e1vel do numerador 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-e074e8c7841e0951ae03d6dfd2bfd1b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\left(2x-1\\right)^2-\\left(\\sqrt{4x^2+1}\\right)^2}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"245\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resolvemos a raiz quadrada:<\/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-17beafd120a7fc185e1499671fb4421a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\left(2x-1\\right)^2-\\left(4x^2+1\\right)}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"218\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resolvemos a identidade not\u00e1vel do quadrado de uma diferen\u00e7a:<\/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-4a34cb3941c92a785c11c50ecaa1e438_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{4x^2+1-4x-\\left(4x^2+1\\right)}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"245\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> 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-1a2e8d86f22087e775650d36bf78e719_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{4x^2+1-4x-4x^2-1}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"228\" style=\"vertical-align: -16px;\"><\/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-2f25890bccb1eaa4c7aa7338f3a25f6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\frac{-4x}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"195\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, por fim, calculamos o valor do limite no infinito:<\/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-6986ded778a6220e3ad9d6c6bf873451_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty} \\cfrac{-4x }{2x-1 +\\sqrt{4x^2+1} } = \\cfrac{-4(+\\infty) }{2(+\\infty)+\\sqrt{4(+\\infty)^2} } = \\cfrac{-\\infty}{+\\infty} =\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"458\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mesmo que haja um x ao quadrado no denominador, seu grau \u00e9 na verdade 1 porque est\u00e1 dentro de uma raiz:<\/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-0decc88d206f476d332becb025b8eeaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4x^2} =\\sqrt{4}\\cdot \\sqrt{x^2} = \\sqrt{4}\\cdot x^{2\/2} =\\sqrt{4} x^1=\\sqrt{4}x .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"351\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, o resultado da indetermina\u00e7\u00e3o -\u221e\/+\u221e \u00e9 a divis\u00e3o dos coeficientes do x de grau superior, pois o grau do numerador \u00e9 igual ao grau do denominador.<\/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-8eb19af7ca51c14245db81bd6781b881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{-4x}{2x-1+\\sqrt{4x^2+1} }=\\frac{-\\infty}{+\\infty}=\\frac{-4}{2+\\sqrt{4}}=\\frac{-4}{2+2}=\\frac{-4}{4}=\\bm{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"499\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Observe que, como existem dois termos de primeiro grau no denominador<\/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-c973910499b6b5a4828e213dc33f948d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl(2x\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -7px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c623cb17f27418239e3fcf7c2ec09946_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4x^2}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"46\" style=\"vertical-align: -7px;\"><\/p>\n<p> , para resolver a indetermina\u00e7\u00e3o -\u221e\/+\u221e \u00e9 necess\u00e1rio tomar todos os coeficientes dos termos de primeiro grau, ou seja, o<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-da4556c0a02b580047678d308649edf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\"><\/p>\n<p> e a<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-65ddaa07508d3929b6969a5e4e6baddf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\"><\/p>\n<p> de <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e8a851efdbfbb4531c82837d5a61edd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4x^2}.\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -1px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 7<\/h3>\n<p> Calcule o limite quando x se aproxima de 1 da seguinte fun\u00e7\u00e3o com 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-480bb119c1303a7afa394d812b0e7602_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1}\\left(\\frac{1}{1-x}-\\frac{3}{1-x^3}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"173\" style=\"vertical-align: -17px;\"><\/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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Tentando fazer o limite, obtemos o limite indeterminado do infinito menos infinito:<\/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-d11d45ea6681f3645773f6e0df8cce9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1}\\left(\\frac{1}{1-x}-\\frac{3}{1-x^3}\\right)=\\frac{1}{1-1}--\\frac{3}{1-1^3}=\\frac{1}{0}-\\frac{3}{0}=\\bm{\\infty-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"480\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Devemos portanto reduzir as fra\u00e7\u00f5es a um denominador comum, ou por outras palavras, devemos multiplicar o numerador e o denominador de uma fra\u00e7\u00e3o pelo denominador da outra:<\/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-75bf3ffa177f32711c5509ce5fe5992d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x \\to 1}\\left(\\frac{1}{1-x}-\\frac{3}{1-x^3} \\right)=\\\\[5ex]\\displaystyle =\\lim_{x\\to 1}\\left( \\frac{1\\cdot(1-x^3)}{(1-x)\\cdot(1-x^3)}-\\frac{3\\cdot(1-x)}{(1-x^3)\\cdot(1-x)}\\right)=\\\\[5ex]\\displaystyle =\\lim_{x \\to 1}\\left(\\frac{1-x^3}{1-x-x^3+x^4}-\\frac{3-3x}{1-x-x^3+x^4}\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"186\" width=\"382\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E como as duas fra\u00e7\u00f5es agora t\u00eam o mesmo denominador, podemos junt\u00e1-las:<\/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-c381a263e89e5a60ff0e6df9367a8ab1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\left(\\frac{1-x^3}{1-x-x^3+x^4}-\\frac{3-3x}{1-x-x^3+x^4}\\right)=\\lim_{x\\to 1}\\frac{1-x^3-(3-3x)}{1-x-x^3+x^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"517\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> 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-05279cd25d55f5c50edfb5f82929701b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1} \\cfrac{1-x^3-3+3x}{1-x-x^3+x^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -14px;\"><\/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-818107141eb339d788408e23078ddda9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1} \\cfrac{-x^3+3x-2}{x^4-x^3-x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E tentamos resolver o limite novamente:<\/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-9d0a31b51faff7e77e778fba66fdbaa1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\frac{-x^3+3x-2}{x^4-x^3-x+1}=\\frac{-1^3+3\\cdot1-2}{1^4-1^3-1+1}=\\mathbf{\\frac{0}{0}}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"335\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mas encontramos a indetermina\u00e7\u00e3o zero dividida por zero. Devemos, portanto, fatorar os polin\u00f4mios do numerador e do denominador:<\/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-e5b8321a511b5e370abe8844bf9624ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\frac{-x^3+3x-2}{x^4-x^3-x+1}=\\lim_{x \\to 1}\\frac{-(x-1)^2(x+2)}{(x-1)^2(x^2+x+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"369\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora simplificamos a fra\u00e7\u00e3o removendo o fator que se repete no numerador e no denominador:<\/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-ab5629bd2fabeb755da37d3abea335b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\frac{-\\cancel{(x-1)^2}(x+2)}{\\cancel{(x-1)^2}(x^2+x+1)}=\\lim_{x \\to 1}\\frac{-(x+2)}{x^2+x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"329\" style=\"vertical-align: -17px;\"><\/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-dbb1676133fe1e33fb4d18078b945959_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\frac{-(x+2)}{x^2+x+1}=\\frac{-(1+2)}{1^2+1+1}=\\frac{-3}{3}=\\bm{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"316\" style=\"vertical-align: -14px;\"><\/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 como resolver a indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e). Voc\u00ea encontrar\u00e1 exemplos dessa indetermina\u00e7\u00e3o com diversos tipos de fun\u00e7\u00f5es e, al\u00e9m disso, poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo de indetermina\u00e7\u00e3o infinita menos infinita. Resolvendo indetermina\u00e7\u00e3o infinita menos infinita Quando o limite de uma fun\u00e7\u00e3o d\u00e1 infinito menos infinito, significa que \u00e9 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/\"> <span class=\"screen-reader-text\">Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e)<\/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-374","post","type-post","status-publish","format-standard","hentry","category-limites-de-funcao"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e) - Matoridade<\/title>\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\/indeterminacao-infinito-menos-infinito-\u221e-\u221e\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e) - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Neste artigo explicamos como resolver a indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e). Voc\u00ea encontrar\u00e1 exemplos dessa indetermina\u00e7\u00e3o com diversos tipos de fun\u00e7\u00f5es e, al\u00e9m disso, poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo de indetermina\u00e7\u00e3o infinita menos infinita. Resolvendo indetermina\u00e7\u00e3o infinita menos infinita Quando o limite de uma fun\u00e7\u00e3o d\u00e1 infinito menos infinito, significa que \u00e9 &hellip; Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e) Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-\u221e-\u221e\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-04T04:08:40+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b1bfc56d2079c86e8ad6e1943311b730_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\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e)\",\"datePublished\":\"2023-07-04T04:08:40+00:00\",\"dateModified\":\"2023-07-04T04:08:40+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/\"},\"wordCount\":1413,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Limites de fun\u00e7\u00e3o\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/\",\"url\":\"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/\",\"name\":\"Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e) - Matoridade\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/#website\"},\"datePublished\":\"2023-07-04T04:08:40+00:00\",\"dateModified\":\"2023-07-04T04:08:40+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/#breadcrumb\"},\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e)\"}]},{\"@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":"Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e) - Matoridade","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\/indeterminacao-infinito-menos-infinito-\u221e-\u221e\/","og_locale":"pt_BR","og_type":"article","og_title":"Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e) - Matoridade","og_description":"Neste artigo explicamos como resolver a indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e). Voc\u00ea encontrar\u00e1 exemplos dessa indetermina\u00e7\u00e3o com diversos tipos de fun\u00e7\u00f5es e, al\u00e9m disso, poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo de indetermina\u00e7\u00e3o infinita menos infinita. Resolvendo indetermina\u00e7\u00e3o infinita menos infinita Quando o limite de uma fun\u00e7\u00e3o d\u00e1 infinito menos infinito, significa que \u00e9 &hellip; Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e) Leia mais &raquo;","og_url":"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-\u221e-\u221e\/","article_published_time":"2023-07-04T04:08:40+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b1bfc56d2079c86e8ad6e1943311b730_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\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e)","datePublished":"2023-07-04T04:08:40+00:00","dateModified":"2023-07-04T04:08:40+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/"},"wordCount":1413,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Limites de fun\u00e7\u00e3o"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/","url":"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/","name":"Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e) - Matoridade","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/#website"},"datePublished":"2023-07-04T04:08:40+00:00","dateModified":"2023-07-04T04:08:40+00:00","breadcrumb":{"@id":"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/#breadcrumb"},"inLanguage":"pt-BR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/pt\/indeterminacao-infinito-menos-infinito-%e2%88%9e-%e2%88%9e\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/pt\/"},{"@type":"ListItem","position":2,"name":"Indetermina\u00e7\u00e3o infinita menos infinita (\u221e-\u221e)"}]},{"@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\/374","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=374"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/374\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/media?parent=374"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/categories?post=374"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/tags?post=374"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}