How to determine the limit $\lim_{x\rightarrow 0} x^\alpha\int_0^x f(t) dt$, for $f \in L^p[0,1]$?

Question

Let $p\geq 1$ and $f\in L^p[0,1]$. For $\alpha\in\mathbb{R}$ determine $\lim_{x\rightarrow 0} x^\alpha\int_0^x f(t) \, dt$.

This problem probably has several cases, but I'm having trouble to determine those cases.