Why does √(x²+x) - x approach 1/2, not 0?

[Moo Of Doom]

I don't really understand what I'm trying to prove..I know f is continuous, therefore all values are finite, as also shown by your theorem. Since f(x) is finite, and so is f(0), the error must also be finite. So to show that it is less than C|x| for some x we just have to choose a really large C, is that somewhat correct even if not rigorous?

You have to use the fact that $f(x)$ is differentiable as well. Take the function $f(x)=x^{1/3}$. All values of this function are finite, but the function is not less than $C|x|$ for any $C$ whatsoever when $x$ is sufficiently close to 0.