I would appreciate help to prove that a twice continuously differentiable convex function from R+ to R has the property that f(x)/x has a limit when x tends to infinity.
(I assume "convex" is in the sense of "smiling", that is that the curve lies above its tangent.)
If f(x) is bounded, the limit is 0.
If f is unbounded, then the limit of f(x) as x goes to infinity is infinity, so you can apply de l'Hospital's rule:
Since f is convex, f'' is always positive hence f' is always increasing; and a function that is always increasing has a limit at infinity.Cheers,