simple limit but I forget how to prove it I have to calculate the following limit 
$$\lim_{x\rightarrow -\infty} \sqrt{x^2+2x+2} - x$$
it is in un undeterminated form. 
I tried to rewrite it as follows:
$$\lim_{x\rightarrow -\infty} \sqrt{x^2+2x+2} - \sqrt{|x|^2}$$
but seems a dead road.
Can anyone suggest a solution?
thanks for your help

Assuming you meant $\sqrt{x^2+2x+2} + x$ (as $\sqrt{x^2+2x+2} - x \to +\infty$ when $x\to-\infty$):
Another option would be to use asymptotics and known Taylor expansions (at $0$): for $x\to-\infty$,
$$
\begin{align*}
\sqrt{x^2+2x+2} + x &= |x|\sqrt{1+\frac{2}{x}+\frac{2}{x^2}} - |x| \\
&= |x|\left( 1+\frac{1}{2}\cdot\frac{2}{x} + o\left(\frac{1}{x}\right) - 1 \right) \\
&= \frac{|x|}{x} + o(1) \sim -1
\end{align*}
$$
so the limit is $-1$.
