{"id":27978,"date":"2024-08-17T13:00:09","date_gmt":"2024-08-17T13:00:09","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27978"},"modified":"2024-08-16T01:01:00","modified_gmt":"2024-08-16T01:01:00","slug":"limites-laterais-definicoes-desenvolvimento-e-exercicios","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/limites-laterais-definicoes-desenvolvimento-e-exercicios\/","title":{"rendered":"Limites Laterais: Defini\u00e7\u00f5es, Desenvolvimento e Exerc\u00edcios"},"content":{"rendered":"<p><center><\/p>\n<h1>Limites Laterais: Defini\u00e7\u00f5es, Desenvolvimento e Exerc\u00edcios<\/h1>\n<p><em><strong>Resumo:<\/strong><br \/>\nEste post explica os limites laterais e bilaterais no c\u00e1lculo, mostrando como s\u00e3o definidos e aplicados tanto de forma intuitiva quanto formal. S\u00e3o apresentados exemplos gr\u00e1ficos e alg\u00e9bricos, discutem-se as condi\u00e7\u00f5es para usar a \u00e1lgebra de limites nesses casos, e incluem-se exerc\u00edcios resolvidos para refor\u00e7ar a compreens\u00e3o. O objetivo \u00e9 oferecer uma vis\u00e3o clara e concisa desses conceitos fundamentais no estudo do c\u00e1lculo.<\/em><\/p>\n<p><strong>Objetivos de Aprendizagem:<\/strong><br \/>\nAo final desta aula, o aluno ser\u00e1 capaz de:<\/p>\n<ul style=\"text-align:left;\">\n<li><strong>Compreender<\/strong> a diferen\u00e7a entre limites laterais e bilaterais.<\/li>\n<li><strong>Definir<\/strong> formalmente os limites laterais pela direita e esquerda.<\/li>\n<li><strong>Aplicar<\/strong> a defini\u00e7\u00e3o de limites laterais em problemas de c\u00e1lculo.<\/li>\n<li><strong>Interpretar<\/strong> graficamente os limites laterais e bilaterais.<\/li>\n<li><strong>Reconhecer<\/strong> as condi\u00e7\u00f5es necess\u00e1rias para que exista um limite bilateral.<\/li>\n<li><strong>Utilizar<\/strong> a \u00e1lgebra de limites em contextos de limites laterais.<\/li>\n<li><strong>Resolver<\/strong> exerc\u00edcios de c\u00e1lculo relacionados aos limites laterais.<\/li>\n<\/ul>\n<p><strong>\u00cdNDICE DE CONTE\u00daDOS:<\/strong><br \/>\n<a href=\"#1\">Introdu\u00e7\u00e3o<\/a><br \/>\n<a href=\"#2\">Ideia Intuitiva de Limites Laterais e Bilaterais<\/a><br \/>\n<a href=\"#3\">Defini\u00e7\u00e3o Formal de Limites Laterais<\/a><br \/>\n<a href=\"#4\">Condi\u00e7\u00f5es para a \u00c1lgebra de Limites<\/a><br \/>\n<a href=\"#5\">Exerc\u00edcios Propostos e Resolvidos<\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/8lkvxEHsF6c\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/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=8lkvxEHsF6c&amp;t=138s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Os limites laterais surgem quando nos deparamos com limites que s\u00f3 poderiam existir pela esquerda ou pela direita,<\/span><\/strong><\/a> mas n\u00e3o de ambos os lados. Os que estudamos at\u00e9 agora s\u00e3o precisamente deste \u00faltimo tipo: para que o limite da fun\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> quando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span> exista, \u00e9 necess\u00e1rio que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> esteja bem definida em ambos os lados de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>; se isso n\u00e3o acontecer, a defini\u00e7\u00e3o de limite n\u00e3o funcionar\u00e1. Como as situa\u00e7\u00f5es em que esse tipo de limites ocorre s\u00e3o frequentes, \u00e9 necess\u00e1rio encontrar uma forma de lidar com elas. Isso \u00e9 resolvido atrav\u00e9s de uma defini\u00e7\u00e3o formal.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Ideia Intuitiva de Limites Laterais e Bilaterais<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=279s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Para que o limite de uma fun\u00e7\u00e3o exista<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span>, quando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span>, \u00e9 necess\u00e1rio que a fun\u00e7\u00e3o esteja bem definida em ambos os lados de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>. Se isso ocorrer, ent\u00e3o falamos de um limite bilateral. E se tal limite tamb\u00e9m resultar em <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">L<\/span><\/span>, ent\u00e3o n\u00e3o haveria problema em escrever<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Agora, imagine que redefinimos essa fun\u00e7\u00e3o de tal forma que seu dom\u00ednio inclua apenas valores maiores que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>. Se fizermos isso, notaremos que o limite deixou de existir (porque existir\u00e3o valores de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> para os quais n\u00e3o far\u00e1 sentido); no entanto, graficamente ainda poder\u00edamos dizer que, quando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> continua tendendo a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">L<\/span><\/span>. A ideia intuitiva que estamos produzindo aqui \u00e9 a do limite lateral pela direita, que \u00e9 o que representar\u00edamos atrav\u00e9s da nota\u00e7\u00e3o<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0^+}f(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">e de forma completamente an\u00e1loga teremos o limite pela esquerda<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0^-}f(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Finalmente, o limite bilateral existir\u00e1 sempre que os limites laterais existirem e forem iguais<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0^-}f(x) = \\lim_{x\\to x_0}f(x) = \\lim_{x\\to x_0^-}f(x)<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Defini\u00e7\u00e3o Formal de Limites Laterais<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=438s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Para definir formalmente os limites laterais, \u00e9 suficiente<\/span> <\/strong><\/a>aplicar uma pequena modifica\u00e7\u00e3o na defini\u00e7\u00e3o original de limite.<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} f(x) = L := \\left(\\forall \\epsilon\\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\exist \\delta \\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\right.<\/span><\/span> <span style=\"background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\lt|x-x_0|\\lt\\delta <\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\rightarrow |f(x)-L|\\lt\\epsilon \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Para os limites laterais pela direita, a defini\u00e7\u00e3o fica assim:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall \\epsilon\\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\exist \\delta \\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\right.<\/span><\/span> <span style=\"background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0 \\lt x \\lt x_0 + \\delta <\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\rightarrow |f(x)-L|\\lt\\epsilon \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Para os limites laterais pela esquerda ser\u00e1:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall \\epsilon\\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\exist \\delta \\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\right.<\/span><\/span> <span style=\"background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0 - \\delta \\lt x \\lt x_0<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\rightarrow |f(x)-L|\\lt\\epsilon \\right)<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Condi\u00e7\u00f5es para a \u00c1lgebra de Limites<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=801s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">O interessante em ter essas defini\u00e7\u00f5es \u00e9 que ambas est\u00e3o contidas<\/span><\/strong><\/a> ao mesmo tempo na defini\u00e7\u00e3o usual de limite, e isso \u00e9 importante porque nos isenta de provar novamente todas as propriedades que j\u00e1 provamos para os limites bilaterais. Toda a \u00e1lgebra de limites funcionar\u00e1 tal como vimos em aulas anteriores, desde que os limites envolvidos sejam da mesma natureza (ambos pela esquerda, ou ambos pela direita, nunca misturados), estejam direcionados ao mesmo ponto e existam nesse ponto.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exerc\u00edcios Propostos e Resolvidos<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to {\\frac{1}{2}}^- } \\sqrt{\\dfrac{x+2}{x+1}} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=994s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 1^+} \\sqrt{\\dfrac{x-1}{x+2}} <\/span><\/span><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1058s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"> <strong>[SOLU\u00c7\u00c3O]<\/strong><\/span><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 2^+} \\left(\\dfrac{x}{x+1} \\right) \\left(\\dfrac{2x+5}{x^2+x} \\right)<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1058s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[SOLU\u00c7\u00c3O]<\/strong><\/span><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 1^-} \\left(\\dfrac{1}{x+1} \\right) \\left(\\dfrac{x+6}{x} \\right) \\left(\\dfrac{3-x}{x} \\right) <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1102s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[SOLU\u00c7\u00c3O]<\/strong><\/span><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{h\\to 0^+ } \\dfrac{\\sqrt{h^2 + 4h +5} - \\sqrt{5}}{h} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1168s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[SOLU\u00c7\u00c3O]<\/strong><\/span><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{h\\to 0^-} \\dfrac{\\sqrt{6} - \\sqrt{5h^2 + 11h +6}}{h}<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1276s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td>a. <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -2^+} (x+3)\\dfrac{|x+2|}{x+2}\n\n<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>b. <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -2^-} (x+3)\\dfrac{|x+2|}{x+2}\n\n<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1378s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<li>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td>a. <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 1^+} \\dfrac{\\sqrt{2x}(x-1)}{|x-1|}\n\n<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>b. <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 1^-}\\dfrac{\\sqrt{2x}(x-1)}{|x-1|}\n\n<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1655s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLU\u00c7\u00c3O]<\/span><\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Limites Laterais: Defini\u00e7\u00f5es, Desenvolvimento e Exerc\u00edcios Resumo: Este post explica os limites laterais e bilaterais no c\u00e1lculo, mostrando como s\u00e3o definidos e aplicados tanto de forma intuitiva quanto formal. S\u00e3o apresentados exemplos gr\u00e1ficos e alg\u00e9bricos, discutem-se as condi\u00e7\u00f5es para usar a \u00e1lgebra de limites nesses casos, e incluem-se exerc\u00edcios resolvidos para refor\u00e7ar a compreens\u00e3o. O [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27972,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":145,"footnotes":""},"categories":[856,571],"tags":[],"class_list":["post-27978","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Limites Laterais: Defini\u00e7\u00f5es, Desenvolvimento e Exerc\u00edcios - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explore uma explica\u00e7\u00e3o detalhada dos limites laterais e bilaterais com defini\u00e7\u00f5es formais, exemplos pr\u00e1ticos e exerc\u00edcios resolvidos\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/pt\/limites-laterais-definicoes-desenvolvimento-e-exercicios\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Limites Laterais: Defini\u00e7\u00f5es, Desenvolvimento e Exerc\u00edcios\" \/>\n<meta property=\"og:description\" content=\"Explore uma explica\u00e7\u00e3o detalhada dos limites laterais e bilaterais com defini\u00e7\u00f5es formais, exemplos pr\u00e1ticos e exerc\u00edcios resolvidos\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/pt\/limites-laterais-definicoes-desenvolvimento-e-exercicios\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2024-08-17T13:00:09+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/limiteslaterales.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Limites Laterais: Defini\u00e7\u00f5es, Desenvolvimento e Exerc\u00edcios\" \/>\n<meta name=\"twitter:description\" content=\"Explore uma explica\u00e7\u00e3o detalhada dos limites laterais e bilaterais com defini\u00e7\u00f5es formais, exemplos pr\u00e1ticos e exerc\u00edcios resolvidos\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/limiteslaterales.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/limites-laterais-definicoes-desenvolvimento-e-exercicios\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/limites-laterais-definicoes-desenvolvimento-e-exercicios\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Limites Laterais: Defini\u00e7\u00f5es, Desenvolvimento e Exerc\u00edcios\",\"datePublished\":\"2024-08-17T13:00:09+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/limites-laterais-definicoes-desenvolvimento-e-exercicios\\\/\"},\"wordCount\":964,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/limites-laterais-definicoes-desenvolvimento-e-exercicios\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/08\\\/limiteslaterales.jpg\",\"articleSection\":[\"C\u00e1lculo Diferencial\",\"Matem\u00e1tica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/limites-laterais-definicoes-desenvolvimento-e-exercicios\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/limites-laterais-definicoes-desenvolvimento-e-exercicios\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/limites-laterais-definicoes-desenvolvimento-e-exercicios\\\/\",\"name\":\"Limites Laterais: Defini\u00e7\u00f5es, Desenvolvimento e Exerc\u00edcios - 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