Evaluating Limits - Finding Multiple Results Only one of which is Correct When I calculate the limit 
$$\lim_{x\to\infty}\sqrt{x^2+x} - \sqrt {x^2-x}$$
I get $2$ answers for this question: $1$ and $0$ but $1$ is the right answer. I don't know why this is the case, however. If you multiply by the conjugate divided by the conjugate (1), you take the radical out of top and get it in the bottom and then if you factor out $x$ from both and cancel it with top u get $2/2$ which is $1$. But if you just factor you get 
$$\lim_{x\to\infty}  x (\sqrt {1+ 1/x} - \sqrt{1 - 1/x}.$$
This simplifies to 0. So how would you know which method to use if you didn't know the right answer?

You are right when multiplying the conjugate
$$\sqrt{x^2+x} - \sqrt {x^2-x}={2x\over\sqrt{x^2+x} + \sqrt {x^2-x}}={2\over\sqrt{1+{1\over x}} + \sqrt {1-{1\over x}}}$$
