Every two asymptotically distinct functions $\mathbb{R}\to\mathbb{R}$ has another function $\mathbb{R}\to\mathbb{R}$ asymptotically between them If $\lim_{x\to\infty}\frac{f(x)}{g(x)}=\infty$ then exists $h(x)$ such that $\lim_{x\to\infty}\frac{h(x)}{g(x)}=\infty$ and $\lim_{x\to\infty}\frac{f(x)}{h(x)}=\infty$
Is this statement true?
I have tried proving this more specific theorem but it doesn't work.
I tried looking for a counter example, with say the slowest growing function as $f(x)$ and $g(x)=1$ but the problem now is a similar one, "is there a slowest growing function?" i.e is there a function $f(x)$ such that $\lim_{x\to\infty}f(x)=\infty$ but for every $g:\mathbb{R}\to\mathbb{R}$ such that $\lim_{x\to\infty}g(x)=\infty$ we have $\limsup_{x\to\infty}\frac{f(x)}{g(x)}<\infty$?
Any help would be appreciated.

Set
$$
h(x) := g(x)\log\left(\frac{f(x)}{g(x)}\right)
$$
for sufficiently large $x$, then we have
$$
\lim_{x\to\infty}\frac{h(x)}{g(x)} = \lim_{x\to\infty} \log\left(\frac{f(x)}{g(x)}\right) = \infty
$$
and
$$
\lim_{x\to\infty}\frac{f(x)}{h(x)} = \lim_{x\to\infty}\frac{\frac{f(x)}{g(x)}}{\log\left(\frac{f(x)}{g(x)}\right)} = \infty.
$$
