TITLE: Difference of asymptotes at infinity
QUESTION [9 upvotes]: Let $f(x)$ and $g(x)$ be two functions such that $$\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$$ Does it imply:

$\lim_{x\to\infty}\frac{g(x)}{f(x)}=1$
$\lim_{x\to\infty}\left(f(x)-g(x)\right)=0$

To me it seems like both are true, but the second is obviously false: $$\lim_{x\to\infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x}\right)=1/2$$ but why? If both functions are basically the same far enough in the number line why does the limit not approach $0$. I don't know how to give a rigorous answer (rather than examples)
Are the answers different in these two cases?

$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}g(x)=\infty$
$f(x)$ and $g(x)$ are bounded

Thanks

REPLY [6 votes]: Note that $f(x)-g(x)=f(x)\left(1-\frac{g(x)}{f(x)}\right)$. Although $\lim_{x\to\infty}\left(1-\frac{f(x)}{g(x)}\right)=0$, but if $\lim_{x\to\infty}f(x)=\infty$, we cannot tell whether $\left[f(x)\left(1-\frac{g(x)}{f(x)}\right)\right]$ tends to $0$ or not.