{"id":29340,"date":"2024-10-22T13:00:19","date_gmt":"2024-10-22T13:00:19","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29340"},"modified":"2024-10-23T14:36:21","modified_gmt":"2024-10-23T14:36:21","slug":"limite-no-infinito-definicoes-e-exemplos","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/limite-no-infinito-definicoes-e-exemplos\/","title":{"rendered":"Limite no Infinito: Defini\u00e7\u00f5es e Exemplos"},"content":{"rendered":"<p><center><\/p>\n<h1>Limite no Infinito: Defini\u00e7\u00f5es e Exemplos<\/h1>\n<p><em><\/p>\n<p><strong>Resumo:<\/strong><br \/>\nEsta aula abordar\u00e1 os limites no infinito, descrevendo o comportamento de <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> quando <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tende ao infinito. Limites b\u00e1sicos como <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to \\infty} \\frac{1}{x} = 0<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to \\infty} k = k<\/span> s\u00e3o explicados, junto com propriedades alg\u00e9bricas semelhantes \u00e0s dos limites finitos.\n<\/p>\n<p><\/em><\/p>\n<p><strong>Objetivos de Aprendizagem:<\/strong><br \/>\nAo final desta aula, o estudante ser\u00e1 capaz de<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Descrever<\/strong> o comportamento de <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> quando <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tende ao infinito.<\/li>\n<li><strong>Definir<\/strong> o limite no infinito usando nota\u00e7\u00e3o matem\u00e1tica formal.<\/li>\n<li><strong>Aplicar<\/strong> propriedades alg\u00e9bricas no c\u00e1lculo de limites no infinito.<\/li>\n<li><strong>Distinguir<\/strong> entre diferentes casos de limites em fun\u00e7\u00f5es racionais no infinito.<\/li>\n<li><strong>Demonstrar<\/strong> a validade das propriedades de soma, subtra\u00e7\u00e3o, multiplica\u00e7\u00e3o, divis\u00e3o e pot\u00eancias de limites no infinito.<\/li>\n<li><strong>Resolver<\/strong> exerc\u00edcios pr\u00e1ticos de limites no infinito em diferentes fun\u00e7\u00f5es.<\/li>\n<\/ol>\n<p><u>\u00cdNDICE DE CONTE\u00daDOS<\/u>:<br \/>\n<a href=\"#1\">Introdu\u00e7\u00e3o<\/a><br \/>\n<a href=\"#2\">Defini\u00e7\u00e3o de Limite no Infinito<\/a><br \/>\n<a href=\"#3\">Limites B\u00e1sicos no Infinito<\/a><br \/>\n<a href=\"#4\">\u00c1lgebra de Limites no Infinito<\/a><br \/>\n<a href=\"#5\">Limite no Infinito em Fun\u00e7\u00f5es Racionais<\/a><br \/>\n<a href=\"#6\">Exemplos de Limites no Infinito<\/a><br \/>\n<\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/MjjSAQLeNBE\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Introdu\u00e7\u00e3o<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=41s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Um dos elementos mais caracter\u00edsticos do c\u00e1lculo \u00e9 o infinito e o limite no infinito.<\/span><\/strong><\/a> O conceito de infinito n\u00e3o se refere a um n\u00famero real; em vez disso, tenta descrever uma magnitude que excede qualquer limite real. Por exemplo, quando temos a fun\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = 1\/x<\/span> e perguntamos sobre seu comportamento quando <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> se torna t\u00e3o grande quanto desejado, quando <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tende ao infinito (<span class=\"katex-eq\" data-katex-display=\"false\">x\\to \\infty<\/span>), observamos que <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> pode, consequentemente, aproximar-se de zero tanto quanto desejado. Assim, escrevemos:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to + \\infty}\\dfrac{1}{x} = 0<\/span>\n<p style=\"text-align: justify; color: #000000;\">Graficamente, essa situa\u00e7\u00e3o se apresenta da seguinte forma:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-fX0lg2ICTTU\/YGD0hNJWI6I\/AAAAAAAAEwQ\/0v8hW6ARkQYDIzT_eG5WVgZ0-pPwwPBwgCLcBGAsYHQ\/s0\/limiteAlinfinito.PNG\" alt=\"limite no infinito\" class=\"alignnone size-full lazyload\" width=\"400\" height=\"300\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-fX0lg2ICTTU\/YGD0hNJWI6I\/AAAAAAAAEwQ\/0v8hW6ARkQYDIzT_eG5WVgZ0-pPwwPBwgCLcBGAsYHQ\/s0\/limiteAlinfinito.PNG\" alt=\"limite no infinito\" class=\"alignnone size-full lazyload\" width=\"400\" height=\"300\" \/><\/noscript><\/center><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Defini\u00e7\u00e3o de Limite no Infinito<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=144s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">A partir desta ideia que acabamos de introduzir<\/span><\/strong><\/a>, podemos formular a defini\u00e7\u00e3o matem\u00e1tica de limite no infinito:<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}f(x) = L := (\\forall\\epsilon\\gt 0) (\\exists M\\in\\mathbb{R})(M\\lt x \\rightarrow |f(x) - L|\\lt \\epsilon )<\/span>\n<p style=\"text-align: justify; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -\\infty}f(x) = L := (\\forall\\epsilon\\gt 0) (\\exists N\\in\\mathbb{R})(x\\lt N \\rightarrow |f(x) - L|\\lt \\epsilon )<\/span>\n<p style=\"text-align: justify; color: #000000;\">A no\u00e7\u00e3o intuitiva deste limite indica o que acontece com <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> quando <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> se afasta tanto quanto desejado da origem, seja indo para a direita ou para a esquerda. A estrat\u00e9gia para o c\u00e1lculo de limites no infinito n\u00e3o \u00e9 muito diferente da que usamos para calcular limites finitos, pois sua \u00e1lgebra \u00e9 praticamente a mesma, apenas precisamos considerar os seguintes resultados:<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Limites B\u00e1sicos no Infinito<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=450s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Com base nestas defini\u00e7\u00f5es, podemos demonstrar<\/span><\/strong><\/a> os seguintes limites b\u00e1sicos.<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}k = k <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}\\dfrac{1}{x} = 0 <\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\">DEMONSTRA\u00c7\u00c3O:<\/span><\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>Pela defini\u00e7\u00e3o de limite no infinito, temos que <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}k = k <\/span> \u00e9 equivalente a dizer: <span class=\"katex-eq\" data-katex-display=\"false\">(\\forall\\epsilon\\gt 0) (\\exists M\\in\\mathbb{R})\\left(M\\lt x \\rightarrow \\left|k-k\\right|\\lt \\epsilon \\right)<\/span>. Mas <span class=\"katex-eq\" data-katex-display=\"false\">\\left|k-k\\right|=0\\lt \\epsilon <\/span> sempre se mant\u00e9m para qualquer <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon \\gt 0,<\/span> e independentemente do valor de <span class=\"katex-eq\" data-katex-display=\"false\">M,<\/span> ent\u00e3o o limite est\u00e1 garantido.\n<p>&nbsp;<\/li>\n<li>Sabe-se que, por defini\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}k = k <\/span> \u00e9 equivalente a dizer: <span class=\"katex-eq\" data-katex-display=\"false\">(\\forall\\epsilon\\gt 0) (\\exists M\\in\\mathbb{R})\\left(M\\lt x \\rightarrow \\left|\\dfrac{1}{x}\\right|\\lt \\epsilon \\right)<\/span>. Mas essa implica\u00e7\u00e3o \u00e9 imediatamente satisfeita se considerarmos <span class=\"katex-eq\" data-katex-display=\"false\">M=1\/\\epsilon,<\/span> de modo que o limite est\u00e1 garantido.\n<p>&nbsp;<\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">Essas demonstra\u00e7\u00f5es s\u00e3o realizadas de forma an\u00e1loga para quando <span class=\"katex-eq\" data-katex-display=\"false\">x\\to+\\infty.<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h2>\u00c1lgebra de Limites no Infinito<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=620s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">A \u00e1lgebra dos limites no infinito \u00e9 an\u00e1loga \u00e0 dos limites finitos.<\/span> <\/strong><\/a>Se <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm \\infty}f(x) = L<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm \\infty}g(x) = M,<\/span> ent\u00e3o as seguintes regras se aplicam:<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><strong>Soma e Subtra\u00e7\u00e3o de Limites:<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}(f(x)\\pm g(x)) = L \\pm M<\/span><\/li>\n<li><strong>Multiplica\u00e7\u00e3o por uma constante:<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}cf(x) = cL<\/span><\/li>\n<li><strong>Produto de Limites:<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}f(x)g(x) = LM<\/span><\/li>\n<li><strong>Divis\u00e3o de Limites:<\/strong> Desde que <span class=\"katex-eq\" data-katex-display=\"false\">M\\neq 0,<\/span> ent\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}f(x)\/g(x)=L\/M<\/span><\/li>\n<li><strong>Pot\u00eancias de Limites:<\/strong> Se <span class=\"katex-eq\" data-katex-display=\"false\">p,q \\in\\mathbb{Z}<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">q\\neq 0<\/span>, ent\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}[f(x)]^{p\/q} = L^{p\/q}<\/span>. Se <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> for par, assume-se que <span class=\"katex-eq\" data-katex-display=\"false\">L\\geq 0<\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">Na verdade, a prova de todas essas propriedades \u00e9 an\u00e1loga \u00e0 dos <a href=\"https:\/\/toposuranos.com\/la-definicion-de-limite-demostraciones-y-teoremas\/\" target=\"_blank\" rel=\"noopener\">limites finitos<\/a><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Limite no Infinito em Fun\u00e7\u00f5es Racionais<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=792s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Uma fun\u00e7\u00e3o racional \u00e9 aquela que pode ser expressa como quociente de dois polin\u00f4mios.<\/span><\/strong><\/a> Ao realizar o c\u00e1lculo de limites no infinito sobre esse tipo de fun\u00e7\u00e3o, podemos observar uma propriedade que \u00e9 muito \u00fatil:<\/p>\n<p style=\"text-align: justify; color: #000000;\">Suponhamos que queremos calcular <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\infty}P(x)\/Q(x)<\/span>\n<ul style=\"text-align: justify; color: #000000;\">\n<li>Se o grau de <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u00e9 maior que o de <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span>, ent\u00e3o o tamanho da fun\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> crescer\u00e1 sem limite quando <span class=\"katex-eq\" data-katex-display=\"false\">x\\to\\infty<\/span> (o limite n\u00e3o existir\u00e1).<\/li>\n<li>Quando o grau de <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u00e9 menor que o de <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span>, ent\u00e3o o limite ser\u00e1 zero.<\/li>\n<li>E, finalmente, se o grau de <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u00e9 igual ao de <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span>, ent\u00e3o o limite ser\u00e1 igual ao quociente dos coeficientes que acompanham o termo de maior grau.<\/li>\n<\/ul>\n<p style=\"text-align: justify; color: #000000;\">O melhor desse resultado \u00e9 que, como veremos nos exemplos a seguir, funciona de modo semelhante mesmo que as pot\u00eancias envolvidas n\u00e3o sejam n\u00fameros inteiros.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Exemplos de Limites no Infinito<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{x+1}{x^2+3}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=907s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -\\infty}\\dfrac{2x^3 + 7}{x^3 - x^2 + x + 7}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=986s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{9x^4 + x}{2x^4 + 5x^2 - x + 6}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1049s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{10x^5 + x4 + 31}{x^4 - 7x^3 + 7x^2 + 9}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1111s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{2\\sqrt{x}+x^{-1}}{3x - 7}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1220s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -\\infty}\\dfrac{2x^{5\/3} - x^{1\/3} + 7}{x^{8\/5}+3x + \\sqrt{x}}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1284s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{\\sqrt[3]{x}-5x+3}{2x + x^{2\/3} - 4}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1406s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{x^{8\/3}+2x + \\sqrt{x}}{x^2+x-3}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1521s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Limite no Infinito: Defini\u00e7\u00f5es e Exemplos Resumo: Esta aula abordar\u00e1 os limites no infinito, descrevendo o comportamento de quando tende ao infinito. Limites b\u00e1sicos como e s\u00e3o explicados, junto com propriedades alg\u00e9bricas semelhantes \u00e0s dos limites finitos. Objetivos de Aprendizagem: Ao final desta aula, o estudante ser\u00e1 capaz de Descrever o comportamento de quando tende [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29336,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":179,"footnotes":""},"categories":[856,571],"tags":[],"citadela-post-location":[],"class_list":["post-29340","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculo-diferencial-pt","category-matematica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Limite no Infinito: Defini\u00e7\u00f5es e Exemplos - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"N\u00e3o entender os limites no infinito est\u00e1 te custando pontos! 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