okay, so the question is what is the limit as x-> infinity of (sqrt(x^2+x))-(sqrt(x^2-x))?
I know it is 1, but how do i show work / come to that conclusion?
A nice trick, probably contrary to what has been beaten into your brain since Algebra I. UNrationalize it!!
In other words, multiply by $\displaystyle \frac{\sqrt{x^{2}+x}+\sqrt{x^{2}-x}}{\sqrt{x^{2}+x}+\sqrt{x^{2}-x}}$.
Resolve the numerator and wonderful things should appear. At least there should be somewhere to go with it.
$\displaystyle \displaystyle\lim_{x\to\infty}(\sqrt{x^2+x}-\sqrt{x^2-x})=\lim_{x\to\infty}\frac{2x}{\sqrt{x^2+x}+\sqrt{ x^2-x}}=$
$\displaystyle \displaystyle =\lim_{x\to\infty}\frac{2x}{x\left(\sqrt{1+\frac{1 }{x}}+\sqrt{1-\frac{1}{x}}\right)}=\lim_{x\to\infty}\frac{2}{\sq rt{1+\frac{1}{x}}+\sqrt{1-\frac{1}{x}}}=1$