$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)}$ where $f(x)$ is linear and $g(x)$ is either strictly convex or strictly concave I am currently working on a problem where I am interested in the limit of a ratio of functions,
$$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)}.$$
It is known that $f(x)$ is linearily increasing, e.g., $f(x) = x$, while $g(x)$ is either strictly convex or strictly concave. 
Intuitively, (but possible incorrectly) it seems to me that, for the case when $g(x)$ is strictly convex, the ratio should tend to $0$ in the limit. Is this a well-known result or is there a good way to show this?
For the second case, when $g(x)$ is concave, is it possible to say anything at all about the limit? To me, it seems that it is not, but that more information is required about the limit of $g(x)$ itself to say something conclusive about the limit of the ratio. 
Many thanks,

The limit can be different from $0$. Actually it can be any real number $m$. Consider for example $f(x)=mx$ and $g(x)=x-\arctan(x)$ which is strictly convex for $x>0$ (or $g(x)=x+\arctan(x)$ for a strictly concave one). Then
$$\lim_{x \rightarrow +\infty} \frac{f(x)}{g(x)}=m.$$
It can be also $\pm \infty$. Take $f(x)=\mp x$ and $g(x)=-\arctan(x)$.
