The impact of asymptotic rates in non asymptotic setting

Question

I know that the topic's title is confusing/vague, but the problem is simple. Suppose you have a function, say $t(k)=a(k)/b(k)$, where $a(\cdot),b(\cdot)$ are some functions such that $a(k)=o(k)$ and $b(k)=o(k)$, for some $k>0$. Furthermore, suppose that $k(\alpha)=n^\alpha$, where $n$ is a quantity that grows to infinity, $\alpha<0$, and that $t(k)\rightarrow c$ as $n\rightarrow\infty$. Now, if I pick $k=n^{\alpha}$ and $k'=n^\beta$, with $\beta<\alpha$, what happens for fixed $n$ is that $t(k^{'}) \leq t(k)$, the difference being larger for small values of $n$. I was wondering whether it is possible to quantify the fact that $t(k^{'})$ is approaching the limit faster than $t(k)$. I am stuck with the fact that, asymptotically, $t(k)=O(1)$ and $t(k^{'})=O(1)$ and so the ratio between these two quantities is still $O(1)$.