Bounds and asymptotic equivalence

Question

Let us suppose 3 functions $f$, $g$ and $h$ where

$$f < g \leq h$$

and

$$\lim_{x \to \infty} f(x)/h(x) = 1.$$

Can we conclude that $\lim_{x \to \infty} f(x)/g(x) = 1$ and $\lim_{x \to \infty} g(x)/h(x) = 1$?

Thank you.

Answer 1

Let's assume that $h(x)>0$; I'll leave you to worry about any other possibilities.

This means that $$\frac{f(x)}{h(x)}<\frac{g(x)}{h(x)}\le\frac{h(x)}{h(x)}\le1.$$ The "squeeze rule" means that $g(x)/h(x)\to1$ etc.