{"id":29463,"date":"2024-11-12T13:00:56","date_gmt":"2024-11-12T13:00:56","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29463"},"modified":"2024-11-13T07:41:18","modified_gmt":"2024-11-13T07:41:18","slug":"limites-infinitos-e-divergencia","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/limites-infinitos-e-divergencia\/","title":{"rendered":"Limites Infinitos e Diverg\u00eancia"},"content":{"rendered":"<style>\np {\ntext-align: justify;}\n<\/style>\n<p><center><\/p>\n<h1>Limites Infinitos e Diverg\u00eancia<\/h1>\n<p style=\"text-align:center;\"><em><strong>Resumo:<\/strong><br \/>\nNesta aula, abordaremos os limites infinitos e os diferentes tipos de diverg\u00eancia nos limites, explorando conceitos fundamentais para entender como certas fun\u00e7\u00f5es n\u00e3o convergem para um valor real definido. Revisaremos os limites laterais diferentes, fun\u00e7\u00f5es com oscila\u00e7\u00f5es infinitas e situa\u00e7\u00f5es em que os limites n\u00e3o existem devido a problemas de dom\u00ednio ou crescimento ilimitado.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Objetivos de Aprendizagem:<\/strong><br \/>\nAo final desta aula, o aluno ser\u00e1 capaz de\n<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Definir<\/strong> limites divergentes e reconhecer quando um limite \u00e9 divergente.<\/li>\n<li><strong>Identificar<\/strong> os diferentes tipos de diverg\u00eancia em limites, como limites laterais diferentes e limites infinitos.<\/li>\n<li><strong>Analisar<\/strong> situa\u00e7\u00f5es em que uma fun\u00e7\u00e3o tem problemas de dom\u00ednio e como isso afeta a exist\u00eancia do limite.<\/li>\n<li><strong>Avaliar<\/strong> limites laterais para determinar se s\u00e3o diferentes e o impacto na converg\u00eancia do limite.<\/li>\n<li><strong>Calcular<\/strong> limites infinitos e distinguir entre limites que divergem para o infinito positivo e o negativo.<\/li>\n<\/ol>\n<p style=\"text-align:center;\">\n<u><strong>\u00cdNDICE DE CONTE\u00daDO<\/strong><\/u>:<br \/>\n<a href=\"#1\"><strong>Quando dizemos que um limite \u00e9 divergente?<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Tipos de Diverg\u00eancia nos Limites<\/strong><\/a><br \/>\n<a href=\"#3\">Limites com Problemas de Dom\u00ednio<\/a><br \/>\n<a href=\"#4\">Limites Laterais Diferentes<\/a><br \/>\n<a href=\"#5\">Limites de Fun\u00e7\u00f5es com Oscila\u00e7\u00f5es Infinitas<\/a><br \/>\n<a href=\"#6\">Limites Infinitos<\/a><br \/>\n<a href=\"#7\">Limites Infinitos no Infinito<\/a>\n<\/p>\n<p><\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/SFBMSd0Q7Io\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p>Nesta ocasi\u00e3o, n\u00e3o revisaremos apenas os <strong>Limites Infinitos<\/strong>, mas tamb\u00e9m os <strong>limites divergentes<\/strong> em geral. Os limites divergentes nos falam sobre como uma fun\u00e7\u00e3o parece n\u00e3o convergir, e isso pode ocorrer de v\u00e1rias formas.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Quando dizemos que um limite \u00e9 divergente?<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=106s\" target=\"_blank\" rel=\"noopener\"><strong>Dizemos que um limite \u00e9 divergente quando ele n\u00e3o converge para algum valor real.<\/strong><\/a> Isso, que soa t\u00e3o \u00f3bvio, pode ocorrer de maneiras diferentes:<\/p>\n<ul>\n<li>Quando os limites laterais s\u00e3o diferentes ou inexistentes, os limites bilaterais n\u00e3o existem.<\/li>\n<li>Se a fun\u00e7\u00e3o n\u00e3o est\u00e1 bem definida, cresce sem limites ou oscila infinitamente ao se aproximar do ponto onde o limite \u00e9 calculado, ent\u00e3o o limite lateral n\u00e3o pode existir.<\/li>\n<\/ul>\n<p>Isso pode se aplicar, com suas particularidades, tanto a limites finitos quanto a limites no infinito, e, dependendo do caso, teremos algum tipo de diverg\u00eancia.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Tipos de Diverg\u00eancia nos Limites<\/h2>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Limites com Problemas de Dom\u00ednio<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=210s\" target=\"_blank\" rel=\"noopener\"><strong>Quando tentamos calcular um limite do tipo <\/strong><\/a><span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to x_0}f(x)<\/span> ou <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to +\\infty}f(x),<\/span> esperamos que pelo menos <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> esteja bem definida para valores pr\u00f3ximos de <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> ou para algum intervalo da forma <span class=\"katex-eq\" data-katex-display=\"false\">[a,+\\infty[, <\/span> respectivamente. Se isso n\u00e3o ocorrer, ent\u00e3o nenhuma das duas defini\u00e7\u00f5es de limites poderia sequer fazer sentido; a fun\u00e7\u00e3o n\u00e3o pode \u00abtender\u00bb para algum valor se ela se aproxima por onde nem sequer est\u00e1 definida. Em tais casos, simplesmente escrevemos que o limite n\u00e3o existe: <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to x_0}f(x)=\\cancel{\\exists}<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to +\\infty}f(x)=\\cancel{\\exists},<\/span> conforme o caso. De maneira similar, isso se aplica a limites laterais, e nada mais precisa ser dito sobre esse tipo de situa\u00e7\u00e3o.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Limites Laterais Diferentes<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=180s\" target=\"_blank\" rel=\"noopener\"><strong>Considere uma fun\u00e7\u00e3o do tipo<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x\/|x|<\/span> e calcule o limite quando <span class=\"katex-eq\" data-katex-display=\"false\">x\\to 0<\/span>. A primeira coisa que notamos \u00e9 que<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0^+} f(x) = 1<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0^-} f(x) = -1<\/span>\n<p>Nesse caso, notamos que, embora os limites laterais existam, eles s\u00e3o diferentes. Quando isso ocorre, simplesmente dizemos que o limite (bilateral) n\u00e3o converge e, portanto:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0} f(x) = \\cancel{\\exists}<\/span>\n<p><a name=\"5\"><\/a><\/p>\n<h3>Limites de Fun\u00e7\u00f5es com Oscila\u00e7\u00f5es Infinitas<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=415s\" target=\"_blank\" rel=\"noopener\"><strong>H\u00e1 tamb\u00e9m o caso de fun\u00e7\u00f5es que, em vez de se aproximarem de um certo valor,<\/strong><\/a> come\u00e7am a oscilar dentro de um determinado intervalo. Um exemplo disso seria uma fun\u00e7\u00e3o do tipo <span class=\"katex-eq\" data-katex-display=\"false\">f(x)= \\sin(1\/x)<\/span>. Se observarmos o que acontece com essa fun\u00e7\u00e3o quando <span class=\"katex-eq\" data-katex-display=\"false\">x\\to 0<\/span>, veremos que ela oscila infinitamente.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-vrdcIAmLV50\/YGoC903Q62I\/AAAAAAAAE1M\/6AM7D-RM9lAy9ZrG105MbfCN8ltu6J09ACLcBGAsYHQ\/s0\/sin1sobrex.PNG\" alt=\"f(x) = sin(1\/x)\" class=\"alignnone size-full lazyload\" width=\"856\" height=\"442\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-vrdcIAmLV50\/YGoC903Q62I\/AAAAAAAAE1M\/6AM7D-RM9lAy9ZrG105MbfCN8ltu6J09ACLcBGAsYHQ\/s0\/sin1sobrex.PNG\" alt=\"f(x) = sin(1\/x)\" class=\"alignnone size-full lazyload\" width=\"856\" height=\"442\" \/><\/noscript><\/p>\n<p>Quando coisas semelhantes ocorrem, dizemos que o limite simplesmente n\u00e3o existe.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Limites Infinitos<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=658s\" target=\"_blank\" rel=\"noopener\"><strong>Vamos ver o que acontece com a fun\u00e7\u00e3o<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = 1\/x.<\/span> A primeira coisa que veremos \u00e9 que, quando <span class=\"katex-eq\" data-katex-display=\"false\">x\\to 0<\/span>, o valor de <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> cresce sem limites, mas a maneira como isso ocorre depender\u00e1 de onde o limite \u00e9 calculado. Intuitivamente, escreveremos<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0^+} \\dfrac{1}{x} = +\\infty<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0^-} \\dfrac{1}{x} = -\\infty<\/span>\n<p>Com essa nota\u00e7\u00e3o, n\u00e3o estamos dizendo que o limite existe de alguma forma; estamos indicando a forma como esse limite n\u00e3o existe. Diferentemente dos casos anteriores, em que o limite n\u00e3o existe e n\u00e3o converge para um valor espec\u00edfico; neste caso, ele diverge porque seu valor ultrapassa qualquer n\u00famero real.<\/p>\n<p>O que acabamos de revisar pode ser formalizado atrav\u00e9s das seguintes defini\u00e7\u00f5es:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0^+}f(x) = +\\infty := \\left(\\forall M \\in \\mathbb{R}\\right)\\left( \\exists \\delta \\gt 0 \\right) ( x_0 \\lt x \\lt x_0 + \\delta \\rightarrow M \\lt f(x) )<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0^-}f(x) = +\\infty := \\left(\\forall M \\in \\mathbb{R}\\right)\\left( \\exists \\delta \\gt 0 \\right) ( x_0 - \\delta \\lt x \\lt x_0 \\rightarrow M \\lt f(x) )<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0}f(x) = +\\infty := \\left(\\lim_{x\\to x_0^+}f(x) = +\\infty \\right) \\wedge \\left(\\lim_{x\\to x_0^-}f(x) = +\\infty \\right)<\/span>\n<p>E de maneira an\u00e1loga:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0^+}f(x) = -\\infty := \\left(\\forall m \\in \\mathbb{R}\\right)\\left( \\exists \\delta \\gt 0 \\right) ( x_0 \\lt x \\lt x_0 + \\delta \\rightarrow f(x) \\lt m )<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0^-}f(x) = -\\infty := \\left(\\forall m \\in \\mathbb{R}\\right)\\left( \\exists \\delta \\gt 0 \\right) ( x_0 - \\delta \\lt x \\lt x_0 \\rightarrow f(x) \\lt m )<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0}f(x) = -\\infty := \\left(\\lim_{x\\to x_0^+}f(x) = -\\infty \\right) \\wedge \\left(\\lim_{x\\to x_0^-}f(x) = -\\infty \\right)<\/span>\n<p>\u00c0s vezes tamb\u00e9m se fala de limite que tende ao infinito (sem sinal)<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0}f(x) = \\infty := \\lim_{x\\to x_0}|f(x)| = +\\infty <\/span>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Limites Infinitos no Infinito<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=1147s\" target=\"_blank\" rel=\"noopener\"><strong>De forma semelhante aos limites revisados anteriormente,<\/strong><\/a> \u00e9 poss\u00edvel definir os limites infinitos no infinito. Por exemplo:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to +\\infty}f(x) = +\\infty := \\left(\\forall M \\in \\mathbb{R}\\right)\\left( \\exists N \\in\\mathbb{R} \\right) ( N\\lt x \\rightarrow M \\lt f(x) )<\/span>\n<p>E com isso j\u00e1 vimos todas as formas em que os limites das fun\u00e7\u00f5es podem divergir.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Limites Infinitos e Diverg\u00eancia Resumo: Nesta aula, abordaremos os limites infinitos e os diferentes tipos de diverg\u00eancia nos limites, explorando conceitos fundamentais para entender como certas fun\u00e7\u00f5es n\u00e3o convergem para um valor real definido. Revisaremos os limites laterais diferentes, fun\u00e7\u00f5es com oscila\u00e7\u00f5es infinitas e situa\u00e7\u00f5es em que os limites n\u00e3o existem devido a problemas de [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29458,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":43,"footnotes":""},"categories":[856,571],"tags":[],"citadela-post-location":[],"class_list":["post-29463","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>Limites Infinitos e Diverg\u00eancia - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Aqui, revisaremos como s\u00e3o definidos os limites infinitos e o que \u00e9 a diverg\u00eancia das fun\u00e7\u00f5es. 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