Answer
$\frac{-4}{5}$
Work Step by Step
$\lim\limits_{x \to -4}\frac{\sqrt {x^2+9}-5}{x+4}=\lim\limits_{x \to -4}\frac{\sqrt {x^2+9}-5}{x+4}*\frac{\sqrt {x^2+9}+5}{\sqrt {x^2+9}+5}=\lim\limits_{x \to -4}\frac{x^2+9-25}{(x+4)\sqrt {x^2+9}+5}=\lim\limits_{x \to -4}\frac{x^2-16}{(x+4)(\sqrt {x^2+9}+5)}=\lim\limits_{x \to -4}\frac{(x+4)(x-4)}{(x+4)(\sqrt {x^2+9}+5)}=\lim\limits_{x \to -4}\frac{x-4}{\sqrt {x^2+9}+5}=\frac{-4-4}{\sqrt {(-4)^2+9}+5}=\frac{-8}{\sqrt {16+9}+5}=\frac{-8}{5+5}=\frac{-4}{5}$
