Does a positive real function tending towards $0$ have a corresponding "convex line of best fit" with respect to area?

Question

Let $\ f:\mathbb{R}^+ \to \mathbb{R}^+\ $ be continuous, such that $\ \displaystyle\lim_{x\to\infty} f(x) = 0.\ $ Does there exist a convex function $\ g:\mathbb{R}^+ \to \mathbb{R}^+\ $ such that $\ \displaystyle\lim_{a\to\infty} \int_0^{a} g(x) - f(x)\ dx = 0\ ?\ $

So in some sense, $\ g(x)\ $ is the "line of best fit" of $\ f(x).$