Answer
$-\dfrac{1}{3}$
Work Step by Step
We want to find $\lim\limits_{x \to -1}\dfrac{\sqrt{x^2+8}-3}{x+1}$, but we can't use the quotient rule for limits because the limit of the denominator as $x$ approaches -1 is zero, and if we try to substitute $x=-1$ directly, we get zero in both the numerator and the denominator.
But note that $\dfrac{\sqrt{x^2+8}-3}{x+1}=\left(\dfrac{\sqrt{x^2+8}-3}{x+1}\right)\left(\dfrac{\sqrt{x^2+8}+3}{\sqrt{x^2+8}+3}\right)=\dfrac{x^2+2-9}{(x+1)(\sqrt{x^2+8}+3)}=\dfrac{x^2-1}{(x+1)(\sqrt{x^2+8}+3)}=\dfrac{(x-1)(x+1)}{(x+1)(\sqrt{x^2+8}+3)}=\dfrac{x-1}{\sqrt{x^2+8}+3}$ for all $x \neq -1.$
Thus $\lim\limits_{x \to -1}\dfrac{\sqrt{x^2+8}-3}{x+1}=\lim\limits_{x \to -1}\dfrac{x-1}{\sqrt{x^2+8}+3}=\dfrac{-1-1}{\sqrt{1+8}+3}=\dfrac{-2}{\sqrt{9}+3}=\dfrac{-2}{3+3}=\dfrac{-2}{6}=-\dfrac{1}{3}.$
