Question

\(\lim\limits_{x\rightarrow1}\dfrac{x+2-\sqrt{x+8}}{\sqrt{x+3}+x-3}\)

Answer 1

\(\lim\limits_{x\rightarrow1}\dfrac{x+2-\sqrt{x+8}}{\sqrt{x+3}+x-3}=\lim\limits_{x\rightarrow1}\left(\dfrac{x^2+3x-4}{x+2+\sqrt{x+8}}:\dfrac{-x^2+7x-6}{\sqrt{x+3}+3-x}\right)\\ =\lim\limits_{x\rightarrow1}\left[\dfrac{\left(x-1\right)\left(x+4\right)}{x+2+\sqrt{x+8}}:\dfrac{\left(x-1\right)\left(6-x\right)}{\sqrt{x+3}+3-x}\right]\\ =\lim\limits_{x\rightarrow1}\dfrac{\left(x+4\right)\left(\sqrt{x+3}+3-x\right)}{\left(6-x\right)\left(x+2+\sqrt{x+8}\right)}=\dfrac{2}{3}\)