Question

como resolver esse limite

Answer 1

Boa tarde Lucas!

Solução!
Para resolver o limite basta escreve-lo de forma diferente,com o objetivo de eliminar a indeterminação.

$\displaystyle \lim_{x \to 5} \frac{ \sqrt{x} - \sqrt{5} }{ \sqrt{x+5} - \sqrt{10}}\\\\\\\ \displaystyle \lim_{x \to 5} \frac{ \sqrt{x} }{\sqrt{x+5} }+ \frac{- \sqrt{5} }{- \sqrt{10} }\\\\\\\\\ \displaystyle \lim_{x \to 5} \frac{ \sqrt{x} }{\sqrt{x+5} }+ \frac{\sqrt{5} }{ \sqrt{10} }\\\\\\\\\ \displaystyle \lim_{x \to 5} \sqrt{ \frac{ x}{x+5} }+ \sqrt{ \frac{5}{10} }\\\\\\\ Substituindo!\\\\\\\ \displaystyle \lim_{x \to 5} \sqrt{ \frac{5}{5+5} }+ \sqrt{ \frac{5}{10} }$


$\displaystyle \lim_{x \to 5} \sqrt{ \frac{5}{10} }+ \sqrt{ \frac{5}{10} }\\\\\\\ Simplificando~~ por~~ 5~~as~~frac\~oes !\\\\\\\ \displaystyle \lim_{x \to 5} \sqrt{ \frac{1}{2} }+ \sqrt{ \frac{1}{2} }\\\\\\\ \displaystyle \lim_{x \to 5} \frac{1}{ \sqrt{2} }+ \frac{1}{ \sqrt{2} }\\\\\\\ Racionalizando~~os~~denominadores~~fica~~assim.$

$\displaystyle \lim_{x \to 5} \frac{1}{ \sqrt{2} }\times \frac{\sqrt{2}}{\sqrt{2}} + \frac{\sqrt{2} }{\sqrt{2}} \times \frac{1}{ \sqrt{2} }\\\\\\\ \displaystyle \lim_{x \to 5} \frac{\sqrt{2}}{ \sqrt{4} } + \frac{\sqrt{2}}{ \sqrt{4} }\\\\\\\ \displaystyle \lim_{x \to 5} \frac{\sqrt{2}+ \sqrt{2} }{ \sqrt{4} } \\\\\\\ \displaystyle \lim_{x \to 5} \frac{2(\sqrt{2})}{ 2} \\\\\\\ \displaystyle \lim_{x \to 5} (\sqrt{2})$


$\boxed{Resposta:~~\displaystyle \lim_{x \to 5} \frac{ \sqrt{x} - \sqrt{5} }{ \sqrt{x+5} - \sqrt{10}}= \sqrt{2}}$

Bom dia!
Bons estudos!