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

HINT:
Putting $y=-x,$
$$ \sqrt{x^2+2x+2} - x=\sqrt{y^2-2y+2} +y= \frac{(\sqrt{y^2-2y+2} +y)(\sqrt{y^2-2y+2} -y)}{(\sqrt{y^2-2y+2} -y)}$$
$$=\frac{y^2-2y+2-y^2}{\sqrt{y^2-2y+2} -y}=\frac{-2+\frac2y}{\sqrt{1-\frac2y+\frac2{y^2}}-1}$$
