{"id":24,"date":"2023-09-17T11:06:22","date_gmt":"2023-09-17T11:06:22","guid":{"rendered":"https:\/\/mathority.org\/pt\/continuidade-de-funcao-continua-de-uma-funcao\/"},"modified":"2023-09-17T11:06:22","modified_gmt":"2023-09-17T11:06:22","slug":"continuidade-de-funcao-continua-de-uma-funcao","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/continuidade-de-funcao-continua-de-uma-funcao\/","title":{"rendered":"Fun\u00e7\u00e3o cont\u00ednua (continuidade de uma fun\u00e7\u00e3o)"},"content":{"rendered":"<p>Neste artigo explicamos o que s\u00e3o fun\u00e7\u00f5es cont\u00ednuas e como determinar se uma fun\u00e7\u00e3o \u00e9 cont\u00ednua num ponto ou n\u00e3o. Al\u00e9m disso, voc\u00ea encontrar\u00e1 propriedades de fun\u00e7\u00f5es cont\u00ednuas e an\u00e1lises de continuidade das fun\u00e7\u00f5es mais comuns. Finalmente, voc\u00ea pode praticar com exerc\u00edcios resolvidos sobre a fun\u00e7\u00e3o cont\u00ednua para compreender totalmente o conceito. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-una-funcion-continua\"><\/span> O que \u00e9 uma fun\u00e7\u00e3o cont\u00ednua?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A continuidade de uma fun\u00e7\u00e3o pode ser estudada graficamente. <strong>Uma fun\u00e7\u00e3o cont\u00ednua \u00e9 uma fun\u00e7\u00e3o que pode ser representada em um gr\u00e1fico sem tirar o l\u00e1pis do papel.<\/strong><\/p>\n<p class=\"has-text-align-center has-medium-font-size\"> <u style=\"text-decoration-color:#FF9B28;\"><strong>Fun\u00e7\u00e3o cont\u00ednua<\/strong><\/u> <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/fonction-continue.webp\" alt=\"fun\u00e7\u00e3o cont\u00ednua\" class=\"wp-image-1491\" width=\"354\" height=\"255\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> A fun\u00e7\u00e3o acima \u00e9 cont\u00ednua porque pode ser desenhada de uma s\u00f3 vez, sem levantar a m\u00e3o do papel.<\/p>\n<p> Por outro lado, quando a condi\u00e7\u00e3o de continuidade anterior n\u00e3o \u00e9 colocada numa fun\u00e7\u00e3o, diz-se que \u00e9 uma <strong>fun\u00e7\u00e3o descont\u00ednua<\/strong> .<\/p>\n<p class=\"has-text-align-center has-medium-font-size\"> <u style=\"text-decoration-color:#FF9B28;\"><strong>Fun\u00e7\u00e3o descont\u00ednua<\/strong><\/u> <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/fonction-pointillee.webp\" alt=\"fun\u00e7\u00e3o descont\u00ednua\" class=\"wp-image-1495\" width=\"354\" height=\"262\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> A fun\u00e7\u00e3o anterior \u00e9 descont\u00ednua porque para represent\u00e1-la \u00e9 necess\u00e1rio fazer duas linhas com o l\u00e1pis. Neste caso, a fun\u00e7\u00e3o deixa de ser cont\u00ednua em x=3, dizemos portanto que x=3 \u00e9 um <strong>ponto de descontinuidade<\/strong> .<\/p>\n<p> Al\u00e9m disso, existem tr\u00eas <strong>tipos de descontinuidades<\/strong> : descontinuidade evit\u00e1vel, descontinuidade inevit\u00e1vel de salto finito e descontinuidade inevit\u00e1vel de salto infinito. No link a seguir voc\u00ea pode ver como \u00e9 cada tipo de descontinuidade e o que h\u00e1 de diferente em cada uma delas:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/tipos-de-descontinuidades\/\">tipos de descontinuidades<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"continuidad-de-una-funcion-en-un-punto\"><\/span> Continuidade de uma fun\u00e7\u00e3o em um ponto<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de vermos a apar\u00eancia do gr\u00e1fico de uma fun\u00e7\u00e3o cont\u00ednua, veremos como saber se uma fun\u00e7\u00e3o \u00e9 cont\u00ednua ou n\u00e3o analiticamente.<\/p>\n<p> Matematicamente, <strong>uma fun\u00e7\u00e3o \u00e9 cont\u00ednua em um ponto se as tr\u00eas condi\u00e7\u00f5es a seguir forem atendidas:<\/strong><\/p>\n<ol style=\"color:#FF8A05; font-weight: bold;border:\">\n<li style=\"margin-bottom:15px\"> <span style=\"color:#101010;font-weight: normal;\">A fun\u00e7\u00e3o existe neste ponto, ou seja, existe a imagem do ponto.<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e7292d399b69758819eab74a2aa9afd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\exists \\ f(a)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">Existe o limite da fun\u00e7\u00e3o neste ponto. Portanto, os limites laterais esquerdo e direito da fun\u00e7\u00e3o neste ponto s\u00e3o iguais.<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7c4c9431d93839b2511a56583101db46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to a^-} f(x) = \\lim_{x \\to a^+} f(x)  \\quad \\color{orange}\\bm{\\longrightarrow}\\color{black} \\quad \\exists \\lim_{x \\to a} f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"413\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:15px\"> <span style=\"color:#101010;font-weight: normal;\">A imagem do ponto coincide com o limite da fun\u00e7\u00e3o neste ponto.<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-da2902aa458fe8a9c8234e2be38336c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(a)=\\lim_{x \\to a} f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"123\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/ol>\n<p> Assim, se as tr\u00eas condi\u00e7\u00f5es de continuidade forem satisfeitas em todos os pontos de uma fun\u00e7\u00e3o, a fun\u00e7\u00e3o \u00e9 cont\u00ednua.<\/p>\n<p> Como exemplo, analisaremos a continuidade da seguinte fun\u00e7\u00e3o por partes: <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/continuite-dune-fonction-definie-par-morceaux.webp\" alt=\"continuidade de uma fun\u00e7\u00e3o definida por partes\" class=\"wp-image-201\" width=\"440\" height=\"306\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> Mesmo se voc\u00ea mudar de se\u00e7\u00e3o, no ponto<\/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-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<p> A fun\u00e7\u00e3o \u00e9 cont\u00ednua, pois os limites laterais da fun\u00e7\u00e3o neste ponto s\u00e3o iguais e coincidem mais com o valor da fun\u00e7\u00e3o neste ponto.<\/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-3f659d88902824c6a762a5ebe2d6db14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x \\to -2^-} f(x)=\\lim\\limits_{x \\to -2^+} f(x)= f(-2)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"300\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Por outro lado, a fun\u00e7\u00e3o n\u00e3o \u00e9 cont\u00ednua no ponto<\/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-2145acc2878ed61214887e120f2485b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> porque os dois limites laterais s\u00e3o diferentes e, portanto, o limite da fun\u00e7\u00e3o n\u00e3o existe neste ponto:<\/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-5f9d62d0c5ddcefe8cc4cbe2daaf6d57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lim\\limits_{x \\to 4^-} f(x)=3 \\neq \\lim\\limits_{x \\to 4^+} f(x)= 2\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"240\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Resumindo, a fun\u00e7\u00e3o definida pelas pe\u00e7as \u00e9 cont\u00ednua em todos os n\u00fameros reais, exceto em<\/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-9ad850560a1ff3e3bb4386cf732d0220_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=4,\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\"><\/p>\n<p> onde h\u00e1 uma descontinuidade.<\/p>\n<p> Tamb\u00e9m podemos verificar que a fun\u00e7\u00e3o \u00e9 descont\u00ednua em<\/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-2145acc2878ed61214887e120f2485b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> pois para represent\u00e1-lo graficamente \u00e9 necess\u00e1rio retirar o l\u00e1pis do papel neste momento. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"continuidad-de-funciones-elementales\"><\/span> Continuidade de fun\u00e7\u00f5es elementares<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Certos tipos de fun\u00e7\u00f5es s\u00e3o cont\u00ednuas pelas suas caracter\u00edsticas:<\/p>\n<ul>\n<li> <strong>As fun\u00e7\u00f5es constantes<\/strong> s\u00e3o cont\u00ednuas em todos os n\u00fameros reais.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-252f9958e223f77bf50cb3b92a7c3e35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=k\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>As fun\u00e7\u00f5es polinomiais<\/strong> s\u00e3o cont\u00ednuas para todos os n\u00fameros reais.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3094a01ff9b10acb7c44349cda083025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=a_0+a_1x+a_2x^2+a_3x^3+\\dots+a_nx^n\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"338\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>As fun\u00e7\u00f5es racionais (ou fracion\u00e1rias)<\/strong> s\u00e3o cont\u00ednuas em todos os n\u00fameros reais exceto em valores que anulam o denominador da fra\u00e7\u00e3o, nestes pontos a fun\u00e7\u00e3o apresenta uma descontinuidade.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6cc33f4306e45701e8bc92457779201d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{p(x)}{q(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"93\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>As fun\u00e7\u00f5es exponenciais<\/strong> s\u00e3o cont\u00ednuas sobre todos os n\u00fameros reais:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-73aa6cf687ce5e8c2f9faf41f5614e53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=a^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>As fun\u00e7\u00f5es logar\u00edtmicas<\/strong> s\u00e3o cont\u00ednuas em todos os pontos que tornam seu argumento positivo.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d94badc2befa1e0c57a867970b1040aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_a (x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"112\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> A continuidade das <strong>fun\u00e7\u00f5es irracionais<\/strong> , ou fun\u00e7\u00f5es com ra\u00edzes, depende do \u00edndice do radical (n). Se o \u00edndice for par, estas s\u00e3o fun\u00e7\u00f5es cont\u00ednuas em todos os pontos que tornam o argumento raiz igual ou maior que zero. Mas se o \u00edndice for \u00edmpar, s\u00e3o fun\u00e7\u00f5es cont\u00ednuas em todos os n\u00fameros reais.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ca6c3302aac609e9214e62444cfec2bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\sqrt[n]{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> A continuidade das <strong>fun\u00e7\u00f5es trigonom\u00e9tricas<\/strong> depende do tipo de fun\u00e7\u00e3o. A fun\u00e7\u00e3o seno e a fun\u00e7\u00e3o cosseno s\u00e3o cont\u00ednuas no conjunto dos n\u00fameros reais, mas a fun\u00e7\u00e3o tangente \u00e9 descont\u00ednua nos pontos\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0c1c82b94111254bf0771af43a5b4411_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x=\\frac{\\pi}{2}+k\\pi\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"91\" style=\"vertical-align: -12px;\"><\/p>\n<p> (onde <em>k<\/em> \u00e9 um n\u00famero inteiro). <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a42292efae266c09baade046e8a3cb76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=sen(x)\\qquad f(x)=cos(x)\\qquad f(x)=tg(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"383\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-de-las-funciones-continuas\"><\/span> Propriedades de fun\u00e7\u00f5es cont\u00ednuas<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Sean<\/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-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/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-6a5054ad96fb0978182340eb214bb125_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"32\" style=\"vertical-align: -5px;\"><\/p>\n<p> duas fun\u00e7\u00f5es cont\u00ednuas no ponto<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c20f4d749ca0651113257e1f1ea66b02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=a,\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: -4px;\"><\/p>\n<p> Pr\u00f3ximo:<\/p>\n<ul>\n<li> A <strong>soma de duas fun\u00e7\u00f5es cont\u00ednuas<\/strong> num ponto \u00e9 outra fun\u00e7\u00e3o cont\u00ednua nesse ponto.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c566932a59271b6e33a784bcf81ee749_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)+g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> O <strong>produto de duas fun\u00e7\u00f5es cont\u00ednuas<\/strong> num ponto \u00e9 igual a outra fun\u00e7\u00e3o cont\u00ednua nesse ponto.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d47b381826b8c5e901bf202b6d4b14c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\\cdot g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>A divis\u00e3o de duas fun\u00e7\u00f5es cont\u00ednuas<\/strong> num ponto resulta em outra fun\u00e7\u00e3o cont\u00ednua nesse ponto, desde que esse ponto n\u00e3o cancele a fun\u00e7\u00e3o de divis\u00e3o.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5962b35380c55e0089deeac0a98b01fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{f(x)}{g(x)}\\qquad g(a)\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"135\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<ul>\n<li> A <strong>composi\u00e7\u00e3o de duas fun\u00e7\u00f5es cont\u00ednuas<\/strong> num ponto d\u00e1 origem a uma fun\u00e7\u00e3o cont\u00ednua neste mesmo ponto.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ba04f43c8995b3659a05f370a10485a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\\circ g(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/composicao-de-funcoes-funcao-composta\/\">o que \u00e9 uma fun\u00e7\u00e3o composta?<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-continuidad-de-una-funcion\"><\/span> Exerc\u00edcios resolvidos sobre a continuidade de uma fun\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Encontre as descontinuidades da fun\u00e7\u00e3o mostrada no gr\u00e1fico a seguir. Determine tamb\u00e9m que tipo de descontinuidade \u00e9. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-discontinuites-de-fonctions.webp\" alt=\"exerc\u00edcio resolveu as descontinuidades de fun\u00e7\u00f5es\" class=\"wp-image-1433\" width=\"400\" height=\"344\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> <strong>Nota:<\/strong> para realizar este exerc\u00edcio recomendamos que veja primeiro quais s\u00e3o os diferentes tipos de descontinuidades e como s\u00e3o identificadas. Voc\u00ea pode ver a explica\u00e7\u00e3o no link do princ\u00edpio dos <u style=\"text-decoration-color:#FF9B28;\">tipos de descontinuidade<\/u> . <\/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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para desenhar a fun\u00e7\u00e3o voc\u00ea deve levantar o l\u00e1pis em x=-2, em x=1 e em x=4. A fun\u00e7\u00e3o \u00e9, portanto, descont\u00ednua nestes tr\u00eas pontos.<\/p>\n<p class=\"has-text-align-left\"> Em x=-2, o limite do lado esquerdo \u00e9 +\u221e e o limite do lado direito \u00e9 3. Assim, como um dos limites laterais \u00e9 infinito, a fun\u00e7\u00e3o tem uma inevit\u00e1vel descontinuidade de salto infinito em x=-2.<\/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-eff79ace33659ceb8203e6e27f38fe35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -2^-} f(x) = +\\infty \\ \\neq \\ \\lim_{x \\to -2^+} f(x) = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"296\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O limite da fun\u00e7\u00e3o em x=1 \u00e9 0 e, por outro lado, o valor da fun\u00e7\u00e3o em x=1 \u00e9 igual a 2. A fun\u00e7\u00e3o apresenta portanto uma descontinuidade evit\u00e1vel em x=1. <\/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-3a616cc5210984ffeed3525a219dd53a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1^-} f(x) =   \\lim_{x \\to 1^+} f(x) = 0 \\ \\bm{\\longrightarrow} \\ \\lim_{x \\to 1} f(x) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"350\" 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-61593bab94da601722d6ae7aedfdb172_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1} f(x) =  0 \\neq  f(0) = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"187\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Em x = 4, o limite do lado esquerdo \u00e9 -3 e o limite do lado direito \u00e9 1. Portanto, como os dois limites laterais s\u00e3o diferentes e nenhum deles d\u00e1 infinito, a fun\u00e7\u00e3o inevitavelmente tem uma descontinuidade de salto finita em x =4. <\/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-b758745a987f927350b2d885009c3a87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 4^-} f(x) = -3 \\ \\neq \\ \\lim_{x \\to 4^+} f(x) = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"265\" 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 2<\/h3>\n<p> Determine os pontos nos quais a fun\u00e7\u00e3o mostrada no gr\u00e1fico a seguir \u00e9 descont\u00ednua. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-types-de-discontinuites-d-une-fonction.webp\" alt=\"exerc\u00edcio resolvido sobre os tipos de descontinuidades de uma fun\u00e7\u00e3o\" class=\"wp-image-1435\" width=\"515\" height=\"434\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> No ponto x=6 a fun\u00e7\u00e3o \u00e9 interrompida porque existe um ponto aberto. O limite quando x se aproxima de 6 \u00e9 -1,4 mas f(6)=1. A fun\u00e7\u00e3o, portanto, tem uma descontinuidade evit\u00e1vel em x=6 porque o valor do limite n\u00e3o coincide com o valor 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-33612be383c71fea04c8c886710f7f10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left. \\begin{array}{l} \\displaystyle \\lim_{x \\to 6^-} f(x)=-1,4\\\\[3ex] \\displaystyle \\lim_{x \\to 6^+} f(x)=-1,4 \\end{array} \\right\\} \\bm{\\longrightarrow} \\lim_{x \\to 6} f(x)=-1,4\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"326\" style=\"vertical-align: 0px;\"><\/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-647b9aea6cc8b605d7e8bc7d5e83ee64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 6} f(x)=-1,4 \\neq f(6)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"217\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Em x=-3 os limites laterais n\u00e3o coincidem e nenhum d\u00e1 o infinito. A fun\u00e7\u00e3o, portanto, tem uma inevit\u00e1vel descontinuidade de salto finito em x=-3.<\/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-3f7d2a2c7abbc525adfcc7576b449f1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to -3^-} f(x)=-2 \\neq \\lim_{x \\to -3^+} f(x)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"275\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E finalmente, a fun\u00e7\u00e3o tem uma inevit\u00e1vel descontinuidade de salto infinito em x = 3, uma vez que pelo menos um limite lateral neste ponto resulta 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-89bdb1598b6153cec4e9efce7b0927b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 3^-} f(x)=+\\infty \\qquad \\lim_{x \\to 3^+} f(x)=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"300\" 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 3<\/h3>\n<p> Analise a continuidade 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-59ee18262fd6f3427ebab6ceabb22868_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\frac{2}{x-5}\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"101\" 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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> As fun\u00e7\u00f5es racionais s\u00e3o cont\u00ednuas em todo o seu dom\u00ednio, ou seja, em todos os n\u00fameros reais, exceto nos valores que cancelam o denominador. Portanto, igualamos o denominador da fun\u00e7\u00e3o racional a zero para ver quais pontos n\u00e3o pertencem ao dom\u00ednio: <\/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-845b4cbd750afedb8e09b0ed6a9809c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"73\" style=\"vertical-align: 0px;\"><\/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-8ddab230605c435eb8b7408a736d3e77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=5\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\"><\/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-96a28ea816d10d62e90dd13ad1aa79c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f = \\mathbb{R} - \\{5\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o ser\u00e1, portanto, cont\u00ednua em todos os pontos, exceto x=5.<\/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> Analise a continuidade da seguinte fun\u00e7\u00e3o por partes: <\/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-a18e6289d268e6ea9fe1ee3ea14d31fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} 5x-2 &amp; \\text{si} &amp;  x < 1 \\\\[2ex] x^2+2 &amp; \\text{si} &amp; x \\geq 1 \\end{array} \\right.\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"218\" style=\"vertical-align: 0px;\"><\/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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o tamb\u00e9m \u00e9 cont\u00ednua na primeira se\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-e05d9a74362639e610acff0ab2d83b16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"5x-2\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<p> , como na segunda se\u00e7\u00e3o,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-79baa9dfae33805a8e2362cc182fd867_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+2\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" style=\"vertical-align: -2px;\"><\/p>\n<p> , uma vez que s\u00e3o fun\u00e7\u00f5es polinomiais.<\/p>\n<p class=\"has-text-align-left\"> Portanto, o \u00fanico ponto em que a fun\u00e7\u00e3o pode ser descont\u00ednua \u00e9 o ponto em que a fun\u00e7\u00e3o se quebra por partes. Ent\u00e3o vamos calcular os limites laterais neste ponto: <\/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-3c84a263e22011d5c8d67c9ee977f98e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1^-} f(x) = \\lim_{x \\to 1} (5x-2)=5\\cdot 1-2=\\bm{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"309\" 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-a1d1b6f5e109c06114b9e36af6f7078a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1^+} f(x) = \\lim_{x \\to 1} (x^2+2)=1^2+2=\\bm{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"293\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os dois limites laterais coincidem portanto, o limite da fun\u00e7\u00e3o quando x tende a 1 \u00e9 igual a 3:<\/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-a621c7ef999aae5418c2f71da1da67f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 1^-} f(x) = \\lim_{x \\to 1^+} f(x) = 3 \\ \\bm{\\longrightarrow} \\ \\exists \\lim_{x \\to 1} f(x) = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"360\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Al\u00e9m disso, a imagem de x=1 tamb\u00e9m \u00e9 3:<\/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-61f3e46233f15d71b39af62ae6bfb4eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=1^2+2=\\bm{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"137\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, como o limite da fun\u00e7\u00e3o em x=1 \u00e9 igual \u00e0 imagem do referido ponto, a fun\u00e7\u00e3o \u00e9 cont\u00ednua no ponto x=1. E, portanto, \u00e9 cont\u00ednuo em todos os n\u00fameros reais. <\/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-8cb4559dfddcda3b3c68a5d64eeadcee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(1)=\\lim_{x \\to 1} f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"122\" 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 5<\/h3>\n<p> Estude a continuidade 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-c43021550a231793494acc2b6eea24d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\sqrt{2x+6}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"123\" style=\"vertical-align: -5px;\"><\/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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u00c9 uma fun\u00e7\u00e3o radical com \u00edndice par, portanto a fun\u00e7\u00e3o ser\u00e1 cont\u00ednua enquanto o argumento do radical for maior que 0 (porque a raiz quadrada de um n\u00famero negativo n\u00e3o existe):<\/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-d54da4e00c0cc48d43914e9e34b1c1d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x+6\\ge 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resolvemos a desigualdade: <\/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-7739f2163d650971be01af6f5a93ae25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x\\ge -6\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"66\" style=\"vertical-align: -3px;\"><\/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-db29954e4e5815cd2bf4bcbfe4c53145_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\\ge \\cfrac{-6}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"67\" style=\"vertical-align: -12px;\"><\/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-dfc84780b14328431bcc372ff904772d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\\ge -3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A solu\u00e7\u00e3o consiste em todos os n\u00fameros maiores ou iguais a -3. A fun\u00e7\u00e3o \u00e9, portanto, cont\u00ednua no intervalo de seu dom\u00ednio: <\/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-1d2a6678926adfe02dc663e155ec573f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mathbf{Dom } \\ \\bm{f = [-3,+\\infty) }\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<\/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> Analise a continuidade da seguinte fun\u00e7\u00e3o logar\u00edtmica: <\/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-e2c2073b5212610fcc153bdb47eacccf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\log_3 (-3x+6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"165\" style=\"vertical-align: -5px;\"><\/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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Esta \u00e9 uma fun\u00e7\u00e3o logar\u00edtmica e n\u00e3o existe o logaritmo de um n\u00famero negativo nem o logaritmo de 0. Portanto, a fun\u00e7\u00e3o existir\u00e1 enquanto o argumento do logaritmo for positivo (maior que zero):<\/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-53c23d1b367425e8cfb94fb36235b8f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3x+6>0&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;14&#8243; width=&#8221;95&#8243; style=&#8221;vertical-align: -2px;&#8221;><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resolvemos a desigualdade: <\/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-d0183470903eba7ebd08dfcdbf49287e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3x>-6&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;14&#8243; width=&#8221;78&#8243; style=&#8221;vertical-align: -2px;&#8221;><\/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-9c67175fd587232ef7d927f7dfca725f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x<\\cfrac{-6}{-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"67\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Lembre-se de que ao dividir um n\u00famero negativo pelo outro lado de uma inequa\u00e7\u00e3o, voc\u00ea deve inverter o sinal da inequa\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-88de95669c546ef6d18c28d0579fbe0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x<2\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"42\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A solu\u00e7\u00e3o consiste em todos os n\u00fameros menores que 2. O dom\u00ednio de defini\u00e7\u00e3o da fun\u00e7\u00e3o \u00e9, portanto:<\/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-505dcdf16c80fdf012163a8800d92836_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mathbf{Dom } \\ \\bm{f = (-\\infty,2) }\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A fun\u00e7\u00e3o \u00e9, portanto, cont\u00ednua em todos os pontos do seu dom\u00ednio.<\/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 a continuidade da seguinte 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-66e02ab966f87bc29b3791246fbf8032_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{4x-2}{\\sqrt{-2x-8}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"139\" style=\"vertical-align: -16px;\"><\/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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> No denominador da fra\u00e7\u00e3o temos um radical com \u00edndice par, portanto a fun\u00e7\u00e3o existir\u00e1 sempre que o conte\u00fado da raiz for igual ou maior que zero:<\/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-042efdf7af54fcbec527f0ee83e1b007_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2x-8\\geq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Mas tamb\u00e9m, a raiz est\u00e1 no denominador da fra\u00e7\u00e3o, e o denominador de uma fra\u00e7\u00e3o nunca pode ser 0. Portanto, a fun\u00e7\u00e3o s\u00f3 existir\u00e1 se o conte\u00fado da raiz for estritamente maior que 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-62002e9d71e0718d04d6d9db08d0e04b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2x-8> 0&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;14&#8243; width=&#8221;95&#8243; style=&#8221;vertical-align: -2px;&#8221;><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora resolvemos a desigualdade: <\/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-336543c46271685f7d069786d2cf4e12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2x>8&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;14&#8243; width=&#8221;64&#8243; style=&#8221;vertical-align: -2px;&#8221;><\/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-3e4a2af9f53dd0a356211660f770666e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x<\\cfrac{8}{-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"59\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Lembre-se de que quando mudamos os lados de um n\u00famero negativo multiplicando ou dividindo uma inequa\u00e7\u00e3o, devemos tamb\u00e9m girar o sinal de inequa\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-ee1c9c50b3c39064f9670f06f9bdc8fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x<-4\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"57\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O resultado s\u00e3o todos os n\u00fameros menores que -4. Assim, o dom\u00ednio da fun\u00e7\u00e3o e, portanto, a sua continuidade, s\u00e3o definidos pelo seguinte intervalo: <\/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-00cfb1cb63133c3092854bb810abfce8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mathbf{Dom } \\ \\bm{f = (-\\infty,-4) }\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 8<\/h3>\n<p> Calcule o valor de <em>k<\/em> para que a fun\u00e7\u00e3o seja cont\u00ednua em todo <\/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-6a134e3092861eafea1239dba23bea40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mathbb{R} .\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\"><\/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-f7d8f0d6e0730c139c3baff0989a8fe7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\left\\{ \\begin{array}{lcl} kx-1 &amp; \\text{si} &amp;  x \\leq 2 \\\\[2ex] 3x^2 - 5  &amp; \\text{si} &amp; x > 2 \\end{array} \\right.&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;65&#8243; width=&#8221;225&#8243; style=&#8221;vertical-align: 0px;&#8221;><\/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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para que a fun\u00e7\u00e3o seja cont\u00ednua, os dois limites laterais no ponto de ruptura devem dar o mesmo resultado. Portanto, calculamos primeiro o limite lateral no ponto de ruptura da parte que n\u00e3o possui <em>k<\/em> :<\/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-7eff7d699bc801340c66f4fd95c30b2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 2^+} f(x) = \\lim_{x \\to 2} (3x^2 - 5) = 3\\cdot 2^2-5=\\bm{7}\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"324\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, para que a fun\u00e7\u00e3o por partes seja cont\u00ednua, o outro limite lateral tamb\u00e9m deve ser igual a 7.<\/p>\n<p class=\"has-text-align-left\"> Tentamos calcular o outro limite lateral:<\/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-1fc80cd47e7fd6cf50b2f2de57c4b40f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 2^-} f(x) = \\lim_{x \\to 2} (kx-1) = k\\cdot2 -1= 2k-1\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"350\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, para que a fun\u00e7\u00e3o seja cont\u00ednua, os dois limites laterais de qualquer ponto devem dar o mesmo resultado. Portanto, igualamos a express\u00e3o obtida do limite a 7 (resultado do outro limite lateral).<\/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-8b99f2f508e7c3f7ab4ebaa9f30638a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2k -1= 7\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E finalmente, resolvemos a equa\u00e7\u00e3o resultante para encontrar o valor de <em>k<\/em> : <\/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-c591d3bbba83f012852966634d0d8234_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2k-1 = 7\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"82\" style=\"vertical-align: 0px;\"><\/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-747038396badd9b1d5e6336cf463fe13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2k = 7 +1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"81\" style=\"vertical-align: -2px;\"><\/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-fa5cd6bc10e8ce9c34542635d995323c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2k = 8\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\"><\/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-1009397e0b35416230f2a5a3d307755a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k = \\cfrac{8}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"44\" style=\"vertical-align: -12px;\"><\/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-ffe7f00b80fc8f5c4961250446632d6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{k =4}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/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 o que s\u00e3o fun\u00e7\u00f5es cont\u00ednuas e como determinar se uma fun\u00e7\u00e3o \u00e9 cont\u00ednua num ponto ou n\u00e3o. Al\u00e9m disso, voc\u00ea encontrar\u00e1 propriedades de fun\u00e7\u00f5es cont\u00ednuas e an\u00e1lises de continuidade das fun\u00e7\u00f5es mais comuns. Finalmente, voc\u00ea pode praticar com exerc\u00edcios resolvidos sobre a fun\u00e7\u00e3o cont\u00ednua para compreender totalmente o conceito. O que \u00e9 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/continuidade-de-funcao-continua-de-uma-funcao\/\"> <span class=\"screen-reader-text\">Fun\u00e7\u00e3o cont\u00ednua (continuidade de uma fun\u00e7\u00e3o)<\/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-24","post","type-post","status-publish","format-standard","hentry","category-limites-de-funcao"],"yoast_head":"<!-- This site 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