On Convex Functions and L'Hôpital's Rule

Why is it so hard to find examples of asymptotically equivalent convex functions that diverge when $x$ goes to infinity, and such that their derivatives are not asymptotically equivalent? We investigate on this question, and propose a condition on the function $g$ that allows to deduce $f'\sim g'$ from $f\sim g$ for every convex function $f$, thus reversing L'Hôpital's Rule. Through a set of examples, we show the connections between our result and Karamata's theory on regular variation.