TITLE: How many times can strictly convex functions intersect?
QUESTION [5 upvotes]: Some time ago, I saw a post related to the number of times that two convex (and continuous) functions' graphs can meet. In general, infinitely many times: one can think, for instance, of $g(x):=x^{2}$ and $f(x)=x^{2}+\sin(x)$.
But, if $f,g:[0,+\infty)\longrightarrow [0,+\infty)$ are continuous, strictly convex and 
$$
\lim_{x\rightarrow +\infty}\frac{f(x)}{g(x)}=+\infty
$$
then, at least intuitively, one can state that $f$ and $g$ meet in, at most, two points, that is, the equation $f(x)-g(x)=0$ has, at most, two solutions. What do you think?

REPLY [3 votes]: Following up from my comment, here is a counterexample where both $f$ and $g$ are convex, continuous, have the desired property $\lim_{x\to\infty}f(x)/g(x) = +\infty$, but their graphs share infinitely many common points:
$\hskip2in$ 
You require that the functions are strictly convex and are defined on $[0, +\infty)$, but it is not difficult to modify the above counterexample; take
$$g(x) = x^2,\text{ for} x\geq 0$$
and
$$f(x) = \begin{cases}
g(x), \text{ for } x \in [0,1],\\
x^4, \text{ otherwise}
\end{cases}
$$
Both functions are continuous, strictly convex, their values are nonnegative, and
$$
\lim_{x\to \infty}\frac{f(x)}{g(x)} = \lim_{x\to\infty}\frac{x^4}{x^2} = \infty.
$$