Does $\frac 1 n \int_0^n \frac {g(x)}{h(x)}\,dx\to 0$ if $\int_0^\infty g(x)\,dx$ converges, $\int_0^\infty h(x)\,dx$ diverges and

Question

$h(x)$ is monotone positive on $[0,\infty)$? It seems like it should be the case since $g$ must go down to $0$ faster than $h$, but I can't formalize the argument since $\frac g h$ can oscillate.

This came as a generalization of Find the $\lim_{n \to \infty}\frac{1}{n}\int_{0}^{n}xg(x)\mathrm{d}x$