I want to show that $\lim_{t\to \infty}t f(t) $ exists and the limit is in $(0, \infty)$ for some function $f$. Now I manage to find two bounds for $f$ in which there exists $M, m, R>0$ such that for $t$ big enough:
$$ \frac{m}{t - R} \leq f(t) \leq \frac{M}{t - R}.$$
So if the limit exists then trivially it's in $(0, \infty)$. However, I'm not sure how to prove $tf(t)$ converges. I know that $f$ is a monotone function and have the two bounds above. Does it suffice for the proof or I need to find further results?
(p.s $f$ is the probability of some $2$-Brownian motion hitting a ball of radius $e^{-t}$ centered at origin before exiting some domain $D$. And I have no clue on how it behaves. Any hints on its property will also be much appreciated).