Question

Lim x—> -2 sqrt(3x+10) -2/ x+2

Answer 1

$\displaystyle\lim_{x\to-2}\frac{\sqrt{3x+10}-2}{x+2}$

Rationalize the numerator:

$\dfrac{\sqrt{3x+10}-2}{x+2}\cdot\dfrac{\sqrt{3x+10}+2}{\sqrt{3x+10}+2}=\dfrac{(3x+10)-4}{(x+2)(\sqrt{3x+10}+2)}=\dfrac3{\sqrt{3x+10}+2}$

where the $x+2$ factors in the numerator and denominator cancel. Then the limit is

$\displaystyle\lim_{x\to-2}\frac{\sqrt{3x+10}-2}{x+2}=\lim_{x\to-2}\frac3{\sqrt{3x+10}+2}=\frac3{\sqrt4+2}=\boxed{\frac34}$