Big O of a difference

Question

Assume $f,g$ are such that $$\lim\limits_{n\to\infty}\frac{f(n)}{g(n)}=r\in\mathbb{R}.$$ Is there anything non-trivial we can infer about $$\left|\frac{f}{g}-r\right|$$ in terms of big-O notation, more sharp than $$\left|\frac{f}{g}-r\right|=O\left(\frac{f}{g}\right)=O(1)?$$

Answer 1

No, the convergence can be as slow as you want, so you only can say that the difference is bounded in a neighburhood of $+\infty$, that is to say it is a $O(1)$.