little-o and 3 functions

Question

If we have 3 function $f$, $g$ and $h$ such that :

• $f$ is not $o(g)$
• $f$ is $o(h)$

Can we conclude that $g$ is $o(h)$ ?

i.e is the following true ?

$lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} \neq 0$

and

$lim_{x \rightarrow \infty} \frac{f(x)}{h(x)} = 0 $

implies

$lim_{x \rightarrow \infty} \frac{g(x)}{h(x)} = 0 $

Thanks

Answer 1

In general, it is not true.

If $\displaystyle \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)}$ exists and is not equal to $0$, then it is true. Otherwise, it is not.