Why are the following functions not equivalent I'm trying to find 
$$\lim_{x\to-\infty}\sqrt{x^2 + 2x} - \sqrt{x^2 - 2x}.$$
However, I kept getting $2$ instead of $-2$, so I graphed the function to see what was going on. See the  picture below.
http://puu.sh/k7Y3f/684aeed22b.png
I found the issue in the steps I took, and it boiled down this. To summarize my steps, I rationalized the function, and then factored out $\sqrt{x^2}$ from each term in the denominator (green graph). However as soon as I sqrt those $x^2$'s, I get a different function? What gives? Why is the purple graph different from the green one?

We have
$$
\begin{align}
\sqrt{x^2+2x}-\sqrt{x^2-2x}
&=\frac{4x}{\sqrt{x^2+2x}+\sqrt{x^2-2x}}\\
&=\frac{4\mathrm{sgn}(x)}{\sqrt{1+\frac2x}+\sqrt{1-\frac2x}}\\
\end{align}
$$
Note that $\frac x{|x|}=\mathrm{sgn}(x)$.
