Question

lim \(\dfrac{\sqrt{3\left(x+1\right)}-3}{x-\sqrt{x+2}}\)

x-> 2 

Answer 1

\(=\lim\limits_{x\rightarrow2}\dfrac{\left(3x+3-9\right)\left(x+\sqrt{x+2}\right)}{\left(x^2-x-2\right)\left(\sqrt{3x+3}+3\right)}=\lim\limits_{x\rightarrow2}\dfrac{3\left(x-2\right)\left(x+\sqrt{x+2}\right)}{\left(x-2\right)\left(x+1\right)\left(\sqrt{3x+3}+3\right)}=\dfrac{3\left(2+\sqrt{2+2}\right)}{\left(2+1\right)\left(\sqrt{3.2+3}+3\right)}=\dfrac{2}{3}\)