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

Clearly
$$\lim_{x\rightarrow -\infty} \sqrt{x^2+2x+2} - x=+\infty+\infty=+\infty$$ 
But 
\begin{gather*}\lim_{x\rightarrow +\infty} \sqrt{x^2+2x+2} - x="\infty-\infty"=\\
=\lim_{x\rightarrow +\infty} \frac{(\sqrt{x^2+2x+2} - x)(\sqrt{x^2+2x+2} + x)}{\sqrt{x^2+2x+2} + x}=\lim_{x\rightarrow +\infty} \frac{2x+2}{\sqrt{x^2+2x+2} + x}=\lim_{x\rightarrow +\infty} \frac{2+2/x}{\sqrt{1+2/x+2/x^2} + 1}=1
\end{gather*} 
