Question

Verifique se a função f(x)=x2-16/x-4 é continua emx=4 e calcule o valor do limite nesse ponto?

Answer 1

$\Large\boxed{\begin{array}{l}\underline{\rm D~\!\!efinic_{\!\!,}\tilde ao~de~func_{\!\!,}\tilde ao~cont\acute inua.}\\\sf Uma~func_{\!\!,}\tilde ao~\acute e~cont\acute inua~em~x=a~quando:\\\sf\blacksquare1~f(a)~est\acute a~d~\!\!efinida\\\sf\blacksquare 2~\displaystyle\lim_{ x \to a}f(x)~existe\\\sf\blacksquare 3~\displaystyle\lim_{x \to a}f(x)=f(a)\end{array}}$

$\Large\boxed{\begin{array}{l}\sf f(x)=\dfrac{x^2-16}{x-4}~em~x=4.\\\sf f(4)~n\tilde ao~est\acute a~d~\!\!efinida~pois~f(4)=\dfrac{0}{0}\\\sf~que~\acute e~indeterminac_{\!\!,}\tilde ao~matem\acute atica\\\sf portanto~a~func_{\!\!,}\tilde ao~n\tilde ao~\acute e~cont\acute inua\\\sf em~x=4.\\\displaystyle\sf\lim_{x \to 4}\dfrac{x^2-16}{x-4}\\\\\displaystyle\sf\lim_{x \to 4}\dfrac{\diagup\!\!\!\!\!\!(x-\diagup\!\!\!\!\!\!4)\cdot(x+4)}{\diagup\!\!\!\!\!\!(x-\diagup\!\!\!\!\!4)}=4+4=8\end{array}}$