Question

Lim X---> 3

(√1+x) -2
_______
x-3

Answer 1

$\mathrm{\lim\limits_{x\to3}\dfrac{\sqrt{1+x}-2}{x-3}=\lim\limits_{x\to3}\dfrac{(\sqrt{1+x}-2)(\sqrt{1+x}+2)}{(x-3)(\sqrt{1+x}+2)}=}\\\\\\ \mathrm{=\lim\limits_{x\to3}\dfrac{(\sqrt{1+x})^2-2^2}{(x-3)(\sqrt{1+x}+2)}=\lim\limits_{x\to3}\dfrac{1+x-4}{(x-3)(\sqrt{1+x}+2)}=}\\\\\\ \mathrm{=\lim\limits_{x\to3}\dfrac{x-3}{(x-3)(\sqrt{1+x}+2)}=\lim\limits_{x\to3}\dfrac{1}{\sqrt{1+x}+2}=}\\\\\\ \mathrm{=\dfrac{1}{\sqrt{1+3}+2}=\dfrac{1}{\sqrt{4}+2}=\dfrac{1}{2+2}=\boxed{\mathbf{\dfrac{1}{4}}}}$