Boundedness of $f'(x)/x$ implies uniform continuity of $f(x)/x$ on $(1,\infty)$

Question

Let $f:(1,\infty) \to \mathbb{R}$ be differentiable, define $g, h:(1,\infty) \to \mathbb{R}$ by $g(x)=f'(x)/x$ and $h(x)=f(x)/x$. Suppose $g$ is bounded. Prove that $h$ is uniformly continuous.

I tried by writing $h$ in terms of $f$ and $g$ as: $h(x)=f(x)g(x)/f'(x)$ and then use the fact that $g$ is bounded, meaning $h(x)\le f(x) M/f'(x)$ for some big natural $M$. But then I get stuck.

Answer 1

Since f´(x)/x is bounded then exists M such that ...