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.

Hi Laurent

(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:

the limit is the same as that of (f(x))'/(x)' = f'(x)/1 = f'(x).

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.


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