Question

lim (√1+x) - 2/ x-3 x ⇒ 3

Answer 1

Oi Thaisa :)

$\lim_{x \to 3} \frac{ \sqrt{1+x}-2 }{x-3} \\ \\ \lim_{x \to 3} \frac{ \sqrt{1+x}-2 }{x-3} . \frac{ \sqrt{1+x}+2 }{ \sqrt{1+x}+2 } = \frac{ (\sqrt{1+x})^2-2^2 }{(x-3) (\sqrt{1+x}+2) } \\ \\ \lim_{x \to 3} \frac{1+x-4}{(x-3) (\sqrt{1+x}+2)} \\ \\ \lim_{x \to 3} \frac{x-3}{(x-3) (\sqrt{1+x}+2)} \\ \\\lim_{x \to 3} \frac{1}{\sqrt{1+x}+2} \ \ \ \ \ (substitui \ agora) \\ \\ \lim_{x \to 3} \frac{1}{\sqrt{1+3}+2} \\ \\ \lim_{x \to 3} \frac{1}{4}$

Então:

$\boxed{ \lim_{x \to 3} \frac{ \sqrt{1+x}-2 }{x-3}= \frac{1}{4} }$