Let $f:\mathbf{R}\to \mathbf{R}_{>0}$ be a "nice" function on $\mathbf{R}$ with positive values. (Let's say just continuous for now, but add anything you like.)
Let $a$ be a positive integer. Suppose that $\lim_{t\to 0^+} \frac{f(t)}{t^a}$ exists and equals a positive real number $L$. Can we conclude that there exists some constant $C>0$ such that $f(t) \leq C t^a$ for $t$ small enough? Moreover, can we take $C=L$??
Edited question. I'm only interested in the limit coming from the right.