What am I doing wrong in this limit calculation?

Hello everyone,

I have tried to calculate the following limit of a function, but my result doesn't match the one that my computer gives me.

$\displaystyle \lim \limits_{x\rightarrow-\infty}x(\sqrt{1+x^2}-\sqrt{2+x^2})$

First I multiply by $\displaystyle \frac{\sqrt{1+x^2}+\sqrt{2+x^2}}{\sqrt{1+x^2}+\sqr t{2+x^2}}$, which gives me

$\displaystyle \lim \limits_{x\rightarrow-\infty}\frac{x(1+x^2-2-x^2)} {\sqrt{1+x^2}+\sqrt{2+x^2}}$

Next I simplify the numerator and factor x out of the roots in the denominator

$\displaystyle \lim \limits_{x\rightarrow-\infty}\frac{-x} {x(\sqrt{\frac{1}{x^2}+1}+\sqrt{\frac{2}{x^2}+1})}$

x cancels out, which gives me

$\displaystyle \lim \limits_{x\rightarrow-\infty}\frac{-1} {\sqrt{\frac{1}{x^2}+1}+\sqrt{\frac{2}{x^2}+1}}=-\frac{1}{2}$

My computer algebra system however tells me that the limit is 1/2. It would be -1/2 if x were approaching positive infinity, so I my problem is somehow related to x approaching negative infinity. But what exactly is it that I am doing wrong?