{"id":231,"date":"2019-05-20T12:21:40","date_gmt":"2019-05-20T12:21:40","guid":{"rendered":"http:\/\/localhost\/?p=231"},"modified":"2019-05-20T13:23:18","modified_gmt":"2019-05-20T13:23:18","slug":"exercicios-limites-de-funcoes","status":"publish","type":"post","link":"http:\/\/localhost\/index.php\/2019\/05\/20\/exercicios-limites-de-funcoes\/","title":{"rendered":"Exerc\u00edcios: Limite de fun\u00e7\u00f5es"},"content":{"rendered":"

$\\newcommand{\\N}{\\mathbb N}\\newcommand{\\R}{\\mathbb R}\\newcommand{\\Z}{\\mathbb Z}\\newcommand{\\A}{\\mathcal A}\\newcommand{\\Q}{\\mathbb Q}$
\n1. Determina os limites das seguintes fun\u00e7\u00f5es em todo ponto\u00a0 $a\\in X’$.<\/p>\n

    \n
  1. $f:\\R\\rightarrow \\R$, $f(x)=x^n$ (com $n\\in\\N^+$).<\/li>\n
  2. $f:[0,\\infty)\\rightarrow\\R$, $f(x)=\\sqrt x$.<\/li>\n
  3. $f:\\R\\rightarrow\\R$, $f(x)=[x]$ onde $[x]$ \u00e9 a parte inteira de $x$. Ou seja, $[x]$ \u00e9 o maior inteiro tal que $[x]\\leq x$.<\/li>\n
  4. \u00a0$f:\\R\\rightarrow\\R$, $f(x)=0$ se $x\\in\\Q$, $f(x)=1$ se $x\\in\\R\\setminus\\Q$.<\/li>\n
  5. $f:\\R\\rightarrow\\R$, $f(x)=0$ se $x\\in\\Q$, $f(x)=x$\u00a0se $x\\in\\R\\setminus\\Q$.<\/li>\n
  6. $f:\\R\\setminus\\{0\\}\\rightarrow \\R$, $f(x)=x\/|x|$.<\/li>\n
  7. $f:\\R\\setminus\\{-2\\}\\rightarrow\\R$, $f(x)=(x^2+3x+2)\/(x+2)$.<\/li>\n<\/ol>\n

    2. Seja $f:\\R\\rightarrow\\R$ a fun\u00e7\u00e3o definida por $f(x)=0$ se $x\\in\\R\\setminus\\Q$ e $f(x)=1\/q$ se $x\\in\\Q$ tal que $x=p\/q$ com $\\mbox{mdc}(p,q)=1$ e $q> 0$. Demonstra que $\\lim_{x\\rightarrow a}f(x)=0$ se $a\\in\\R\\setminus\\Q$, enquanto $\\lim_{x\\rightarrow a}f(x)$ n\u00e3o existe se $a\\in Q$. [Dica: veja o argumento no livro de Elon.]<\/p>\n

    3. Demonstre os resultados que foram apresentados na aula, mas as demonstra\u00e7\u00f5es foram omitidas.<\/p>\n

    4. Mostre, exibindo um contraexemplo, que a condi\u00e7\u00e3o que $\\lim_{y\\rightarrow b}g(y)=g(b)$ \u00e9 necess\u00e1ria no resultado sobre o limite da composi\u00e7\u00e3o de fun\u00e7\u00f5es.<\/p>\n

    5. Enuncie e demonstre os resultados sobre limite de fun\u00e7\u00f5es apresentados na aula para limites laterais.<\/p>\n","protected":false},"excerpt":{"rendered":"

    $\\newcommand{\\N}{\\mathbb N}\\newcommand{\\R}{\\mathbb R}\\newcommand{\\Z}{\\mathbb Z}\\newcommand{\\A}{\\mathcal A}\\newcommand{\\Q}{\\mathbb Q}$ 1. Determina os limites das seguintes fun\u00e7\u00f5es em todo ponto\u00a0 $a\\in X’$. $f:\\R\\rightarrow \\R$, $f(x)=x^n$ (com $n\\in\\N^+$). $f:[0,\\infty)\\rightarrow\\R$, $f(x)=\\sqrt x$. $f:\\R\\rightarrow\\R$, $f(x)=[x]$ onde $[x]$ \u00e9 a parte inteira de $x$. Ou seja, $[x]$ \u00e9 o maior inteiro tal que $[x]\\leq x$. \u00a0$f:\\R\\rightarrow\\R$, $f(x)=0$ se $x\\in\\Q$, $f(x)=1$ se $x\\in\\R\\setminus\\Q$. $f:\\R\\rightarrow\\R$, … Continue reading Exerc\u00edcios: Limite de fun\u00e7\u00f5es<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts\/231"}],"collection":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/comments?post=231"}],"version-history":[{"count":6,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts\/231\/revisions"}],"predecessor-version":[{"id":240,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts\/231\/revisions\/240"}],"wp:attachment":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/media?parent=231"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/categories?post=231"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/tags?post=231"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}