Answer
$\lim\limits_{x \to -2}\frac{x+2}{\sqrt(x^{2}+5)-3} = -\frac{3}{2} = -1.5$
Work Step by Step
$\frac{x+2}{\sqrt(x^{2}+5)-3} = \frac{(x+2)(\sqrt(x^{2}+5)+3)}{(\sqrt(x^{2}+5)-3)(\sqrt(x^{2}+5)+3)} = \frac{(x+2)(\sqrt(x^{2}+5)+3)}{(x^{2}+5)-9} = \frac{(x+2)(\sqrt(x^{2}+5)+3)}{(x^{2}-4)}$
= $\frac{(x+2)(\sqrt(x^{2}+5)+3)}{(x+2)(x-2)} =\frac{(\sqrt(x^{2}+5)+3)}{(x-2)}$
Now,
$\lim\limits_{x \to -2}\frac{x+2}{\sqrt(x^{2}+5)-3} = \lim\limits_{x \to -2}\frac{(\sqrt(x^{2}+5)+3)}{(x-2)} = \frac{3+3}{-2-2} =\frac{6}{-4} = -\frac{3}{2} = -1.5$
