Consider two increasing functions $f,g:\mathbb{N}_{>0}\to \mathbb{N}_{>0}$ and let $h:=f/g$ (we assume $g(x)\neq 0$ for all $x>0$). In other words, $f(x)< f(x+1)$, $g(x)< g(x+1)$ holds for all $x>0$ and $h(x)=f(x)/g(x)$.
We assume that $\lim_{x\to \infty}h(x)=0$. Then, is $h$ unimodal (in $x>0$) ? i.e., $\cdots \leq h(x^*-1)\leq h(x^*)\geq h(x^*+1) \geq\cdots$ for some $x^*>0$. For example, $h(x)=x/(x^2+1)$ is unimodal (in $x>0$).
No. Here is a counterexample: $$ f(x)=e^{2x+\sin(x)}\quad g(x)=e^{2x-\sin(x)+\log(1+x^2)},$$ for $x>0$. It is easy to check that both are monotone increasing, and that their ratio is not.