What is wrong with my logic and derivatives? Since the derivative of a function is analogous to a description on how fast the function grows, I thought that $\lim_{x\to\infty}\frac{f'(x)}{f(x)}=0$ for the following reason. Assume $f(x)$ is monotone increasing on an interval $(a,\infty]$. If $f'(x)$ is large, that means that $f(x)$ is getting larger and larger. From this, I've assumed that $f'(x)$ must eventually be outdone by $f(x)$ simply because $f'(x)$ cannot get large without making $f(x)$ even larger. But that's obviously not the case because if $f(x)=\Gamma(x)$, then $\lim_{x\to\infty}\frac{f'(x)}{f(x)}\ne0$. I'm just wondering what is wrong with my logic, that's all.

In short there exist functions such as $e^x$ whose derivative increases at a rate equal to the function itself. You make an incorrect assumption _precisely_ here:

> "From this, I've assumed that $f′(x)$ must eventually be outdone by $f(x)$ simply because $f′(x)$ cannot get large without making $f(x)$ even larger."

With that being said, these sorts of questions are really quite laudable.