Is the following correct?

$\displaystyle \text{Calculate } \lim_{x \rightarrow -1 } \frac{\sqrt{x^2+8}-3}{x+1}$

$\displaystyle = \lim_{x \rightarrow -1 } \frac{\sqrt{x^2+8}-3}{x+1 } \cdot \frac{\sqrt{x^2+8}+3}{\sqrt{x^2+8}+3}$

$\displaystyle = \lim_{x \rightarrow -1 } \frac{x^2+8-9}{(x+1)(\sqrt{x^2+8}+3)}$

$\displaystyle = \lim_{x \rightarrow -1 } \frac{x^2-1}{(x+1)(\sqrt{x^2+8}+3)}$

$\displaystyle = \lim_{x \rightarrow -1 } \frac{(x-1)(x+1)}{(x+1)(\sqrt{x^2+8}+3)}$

$\displaystyle = \lim_{x \rightarrow -1 } \frac{(x-1)}{(\sqrt{x^2+8}+3)}$

$\displaystyle = \frac{(-1-1)}{(\sqrt{1^2+8}+3)}$

$\displaystyle = -\frac{1}{3}$