Question

QUESTAO DE VIDA OU MORTE - Seja f a função representada no gráfico, determine os limites ao lado:

Answer 1

$\large\\\boxed{\begin{array}{l}\sf o~limite~de~uma~func_{\!\!,}\tilde ao~existe\\\sf se~e~somente~se~os~limites~laterais~existem\\\sf e~s\tilde ao~iguais.\\\tt matematicamente:\\\displaystyle\sf\lim_{x \to a}f(x)\exists\Longleftrightarrow \lim_{x \to a^{+}}f(x)=\lim_{x \to a^{-}}f(x)\end{array}}$

$\large\boxed{\begin{array}{l}\tt a)~\displaystyle\sf\lim_{x \to 4^{-}}f(x)=1\\\displaystyle\sf \lim_{x \to 4^{+}}f(x)=1\\\displaystyle\sf como~\lim_{x \to 4^{-}}f(x)=\lim_{x \to 4^{+}}f(x)\\\displaystyle\sf\lim_{x \to 4}f(x)=1\end{array}}$

$\large\boxed{\begin{array}{l}\tt b)~\displaystyle\sf quando~x~decresce~muito,a~func_{\!\!,}\tilde ao~se~aproxima~de~1\\\displaystyle\sf \lim_{x \to -\infty}f(x)=1\end{array}}$

$\boxed{\begin{array}{l}\tt c)~\displaystyle\sf quando~x~cresce~muito,a~func_{\!\!,}\tilde ao~tende~para~menos~infinito\\\sf ou~seja,quando~x~supera~5,a~func_{\!\!,}\tilde ao~tem~imagens~\\\sf negativas.\\\displaystyle\sf assim,\lim_{x \to +\infty}\!\!\!f(x)=-\infty\end{array}}$

$\large\boxed{\begin{array}{l}\tt d)~\displaystyle\sf \lim_{~~x \to 1^{-}}\!\!\!f(x)=0\\\displaystyle\sf\lim_{x \to 1^{+}}f(x)=-1\\\displaystyle\sf\lim_{x \to 1^{-}}f(x)\ne\lim_{x \to 1^{+}}f(x)\implies \not\exists\lim_{x \to 1}f(x)\end{array}}$