{"id":29564,"date":"2024-11-18T17:33:19","date_gmt":"2024-11-18T17:33:19","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29564"},"modified":"2024-11-18T17:33:19","modified_gmt":"2024-11-18T17:33:19","slug":"limite-e-continuidade","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/limite-e-continuidade\/","title":{"rendered":"Limite e Continuidade"},"content":{"rendered":"<style>\np {\ntext-align:justify;\n}\n<\/style>\n<p><center><\/p>\n<h1>Limite e Continuidade<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><em><strong>Resumo:<\/strong><\/em><br \/>\nEsta aula aborda a rela\u00e7\u00e3o entre o limite e a continuidade de uma fun\u00e7\u00e3o, come\u00e7ando com uma explica\u00e7\u00e3o intuitiva e formal do termo. A continuidade em um ponto e em um conjunto \u00e9 explorada, detalhando as condi\u00e7\u00f5es necess\u00e1rias para que uma fun\u00e7\u00e3o seja cont\u00ednua. Tamb\u00e9m s\u00e3o apresentadas as propriedades alg\u00e9bricas das fun\u00e7\u00f5es cont\u00ednuas, incluindo somas, produtos, quocientes e composi\u00e7\u00e3o de fun\u00e7\u00f5es. Finalmente, s\u00e3o resolvidos exemplos pr\u00e1ticos para analisar a continuidade de diferentes fun\u00e7\u00f5es e calcular limites, relacionando-os ao conceito de continuidade.\n<\/p>\n<p style=\"text-align:center;\"><strong>Objetivos de Aprendizagem:<\/strong><br \/>\nAo final desta aula, o aluno ser\u00e1 capaz de:<\/p>\n<ol>\n<li><strong>Analisar<\/strong> a continuidade de uma fun\u00e7\u00e3o em um ponto verificando as condi\u00e7\u00f5es definidas.<\/li>\n<li><strong>Avaliar<\/strong> se uma fun\u00e7\u00e3o \u00e9 cont\u00ednua em um determinado conjunto usando a defini\u00e7\u00e3o formal de continuidade.<\/li>\n<li><strong>Demonstrar<\/strong> a continuidade de uma fun\u00e7\u00e3o a partir de suas propriedades alg\u00e9bricas.<\/li>\n<li><strong>Classificar<\/strong> os pontos de descontinuidade em diferentes tipos de acordo com as condi\u00e7\u00f5es que n\u00e3o s\u00e3o cumpridas.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>\u00cdNDICE DE CONTE\u00daDOS<\/u>:<\/strong><br \/>\n<a href=\"#1\">Introdu\u00e7\u00e3o<\/a><br \/>\n<a href=\"#2\">Continuidade em um Ponto<\/a><br \/>\n<a href=\"#3\">Continuidade em um Conjunto<\/a><br \/>\n<a href=\"#4\">Propriedades das Fun\u00e7\u00f5es Cont\u00ednuas<\/a><br \/>\n<a href=\"#5\">Exerc\u00edcios<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/4kehfJcyPRg\" 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 ><a href=\"https:\/\/www.youtube.com\/watch?v=4kehfJcyPRg&amp;t=109s\" target=\"_blank\" rel=\"noopener\"><strong>Intuitivamente, a continuidade<\/strong><\/a> de uma fun\u00e7\u00e3o \u00e9 entendida como a propriedade que permite desenhar seu gr\u00e1fico \u00absem levantar o l\u00e1pis\u00bb. Esta no\u00e7\u00e3o, que \u00e9 bastante simples de entender por si s\u00f3, n\u00e3o \u00e9 suficiente, no entanto, se o que se deseja \u00e9 o rigor do racioc\u00ednio matem\u00e1tico. \u00c9 necess\u00e1rio um m\u00ednimo de formalidade. Formalmente falando, a continuidade das fun\u00e7\u00f5es est\u00e1 intimamente ligada ao conceito de limite, e \u00e9 isso que estudaremos em detalhe.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>Continuidade em um Ponto<\/h3>\n<p ><a href=\"https:\/\/www.youtube.com\/watch?v=4kehfJcyPRg&amp;t=152s\" target=\"_blank\" rel=\"noopener\"><strong>Uma fun\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u00e9 cont\u00ednua em um ponto<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">x=x_0<\/span> se, e somente se, as seguintes condi\u00e7\u00f5es forem atendidas:<\/p>\n<ul >\n<li><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> est\u00e1 definida em <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/li>\n<li>O limite <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} f(x)<\/span> existe<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} f(x) = f(x_0)<\/span><\/li>\n<\/ul>\n<p >Tudo isso pode ser sintetizado matematicamente atrav\u00e9s da express\u00e3o<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall \\epsilon \\gt 0 \\right) \\left(\\exists \\delta \\gt 0 \\right) \\left(0\\lt |x-x_0|\\lt \\delta \\rightarrow |f(x) - f(x_0)|\\lt \\epsilon\\right)<\/span>\n<p >Esta \u00e9 precisamente a defini\u00e7\u00e3o matem\u00e1tica de limite, em que o valor \u00abL\u00bb foi substitu\u00eddo por \u00abf(<span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span>)\u00bb<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Continuidade em um Conjunto<\/h3>\n<p ><a href=\"https:\/\/www.youtube.com\/watch?v=4kehfJcyPRg&amp;t=462s\" target=\"_blank\" rel=\"noopener\"><strong>A defini\u00e7\u00e3o de continuidade<\/strong><\/a> pontual pode ser estendida para a continuidade em todos os pontos dentro de um certo conjunto. Dizemos que <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u00e9 cont\u00ednua dentro de um conjunto <span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq Dom(f)<\/span> se for cont\u00ednua em todos os pontos <span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in I.<\/span> Isto pode ser sintetizado matematicamente atrav\u00e9s da seguinte express\u00e3o.<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall x_0 \\in I\\subseteq Dom(f) \\right)\\left(\\forall \\epsilon \\gt 0 \\right) \\left(\\exists \\delta \\gt 0 \\right) \\left(0\\lt |x-x_0|\\lt \\delta \\rightarrow |f(x) - f(x_0)|\\lt \\epsilon\\right)<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Propriedades das Fun\u00e7\u00f5es Cont\u00ednuas<\/h3>\n<h4>\u00c1lgebra de Fun\u00e7\u00f5es Cont\u00ednuas<\/h4>\n<p ><a href=\"https:\/\/www.youtube.com\/watch?v=4kehfJcyPRg&amp;t=575s\" target=\"_blank\" rel=\"noopener\"><strong>Se <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> s\u00e3o fun\u00e7\u00f5es cont\u00ednuas<\/strong><\/a> em <span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq Dom(f)\\cap Dom(g)<\/span>, ent\u00e3o teremos que:<\/p>\n<ul >\n<li><span class=\"katex-eq\" data-katex-display=\"false\">f\\pm g<\/span> \u00e9 cont\u00ednua em <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">f\\cdot g<\/span> \u00e9 cont\u00ednua em <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/li>\n<li>Se <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha\\in\\mathbb{R}<\/span>, ent\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha f<\/span> \u00e9 cont\u00ednua em <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">f\/g<\/span> \u00e9 cont\u00ednua em <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> sempre que <span class=\"katex-eq\" data-katex-display=\"false\">g\\neq 0<\/span><\/li>\n<li>Se <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta\\in\\mathbb{R}<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\beta\\neq 0<\/span>, ent\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">f^{\\alpha\/\\beta}<\/span> \u00e9 cont\u00ednua em <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>, desde que <span class=\"katex-eq\" data-katex-display=\"false\">f^{\\alpha\/\\beta}<\/span> esteja bem definida em <span class=\"katex-eq\" data-katex-display=\"false\">I.<\/span><\/li>\n<\/ul>\n<p >Todas essas propriedades que parecem dif\u00edceis de demonstrar a partir da defini\u00e7\u00e3o formal de continuidade s\u00e3o, na verdade, muito f\u00e1ceis, pois s\u00e3o praticamente an\u00e1logas \u00e0 \u00e1lgebra dos limites que <a href=\"https:\/\/toposuranos.com\/la-definicion-de-limite-demostraciones-y-teoremas\/\" rel=\"noopener\" target=\"_blank\">j\u00e1 temos demonstrado<\/a>, pelo que estamos isentos de faz\u00ea-las.<\/p>\n<h5>Composi\u00e7\u00e3o de Fun\u00e7\u00f5es Cont\u00ednuas<\/h5>\n<p >Se <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> \u00e9 cont\u00ednua em <span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq Dom(f)<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u00e9 cont\u00ednua em <span class=\"katex-eq\" data-katex-display=\"false\">f(I)<\/span>, ent\u00e3o a composi\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">(g\\circ f)(x) = g(f(x))<\/span> \u00e9 cont\u00ednua em <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>. Isto, como j\u00e1 se intui, \u00e9 uma consequ\u00eancia direta das leis de composi\u00e7\u00e3o para os limites das fun\u00e7\u00f5es.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exerc\u00edcios<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/R91vswfQT_Q\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<ol >\n<li>Analise a continuidade das seguintes fun\u00e7\u00f5es<br \/>\n<table>\n<tbody>\n<tr>\n<td width=\"30\">a.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{1}{x-2} - 3x<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=33s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>b.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{x+1}{x^2-4x+3}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=97s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>c.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=|x-1| + \\sin(x)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=215s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>d.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{\\cos(x)}{x}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=450s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>e.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=csc(2x)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=564s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>f.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{x\\tan(x)}{x^2 + 1}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=666s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>g.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\sqrt{2x + 3}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=784s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>h.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{x+2}{\\cos(x)}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=917s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>i.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{\\sqrt{x^4 + 1}}{1+\\sin^2(x)}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=952s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Calcule os seguintes limites. As fun\u00e7\u00f5es s\u00e3o cont\u00ednuas nos pontos aos quais se aproximam?<br \/>\n<table>\n<tbody>\n<tr>\n<td width=\"30\">a.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to\\pi} \\sin(x -\\sin(x))<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1033s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>b.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0} \\sin\\left(\\dfrac{\\pi}{2}\\cos(\\tan(x)) \\right)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1092s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>c.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 1} \\sec\\left(x\\sec^2(x) - \\tan^2(x) -1 \\right)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1162s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>d.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0} \\tan\\left(\\dfrac{\\pi}{4}\\cos\\left(\\sin\\left(x^{1\/3}\\right)\\right) \\right)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1274s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>e.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0} \\cos\\left(\\dfrac{\\pi}{\\sqrt{19 - 3\\sec{2x}}} \\right)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1424s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>f.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to \\pi\/6} \\sqrt{\\csc^2(x) + 5\\sqrt{3}\\tan(x)}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1505s\" target=\"_blank\" rel=\"noopener\"><strong>SOLU\u00c7\u00c3O<\/strong><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Limite e Continuidade Resumo: Esta aula aborda a rela\u00e7\u00e3o entre o limite e a continuidade de uma fun\u00e7\u00e3o, come\u00e7ando com uma explica\u00e7\u00e3o intuitiva e formal do termo. A continuidade em um ponto e em um conjunto \u00e9 explorada, detalhando as condi\u00e7\u00f5es necess\u00e1rias para que uma fun\u00e7\u00e3o seja cont\u00ednua. Tamb\u00e9m s\u00e3o apresentadas as propriedades alg\u00e9bricas das [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29557,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":21,"footnotes":""},"categories":[856,571],"tags":[],"class_list":["post-29564","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 v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Limite e Continuidade - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"O que \u00e9 continuidade e qual \u00e9 a rela\u00e7\u00e3o dela com o limite? 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