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?

If you do not like negative numbers, well, just apply a change of variable bringing them into positive numbers:
$$\begin{eqnarray*} \lim_{x\to -\infty}\left(\sqrt{x^2+2x}-\sqrt{x^2-2x}\right)&=&\lim_{z\to +\infty}\left(\sqrt{z^2-2z}-\sqrt{z^2+2z}\right)\\&=&\lim_{z\to +\infty}\frac{-4z}{\sqrt{z^2+2z}+\sqrt{z^2-2z}}=\color{red}{-2}.\end{eqnarray*}$$
