I'm trying to solve the following problem but I'm stuck at one point.
Let $1<p<\infty$ and $f\in L^p[0,\infty)$. Show that $$\lim_{x\to\infty}\cfrac{1}{x^{1-1/p}}\int_0^x f(t)dt=0.$$
And a hint is given: first assume f has compact support.
So far, I've applied Holder's inequality and have got:
$$\left\lvert\cfrac{1}{x^{1-1/p}}\int_0^x f(t)dt\right\rvert\leq\left(\int_0^x|f(t)|^pdt\right)^{1/p}$$
I'm stuck here and unsure how to proceed. How do I make the term on the right arbitrarily small? As $x\to\infty$, it seems the term on the right approaches the $p$-norm of $f$, which has no reason to be $0$. Moreover, I don't know how to use the hint. Any help with this problem is appreciated.
P.S. Interestingly, I first misread the problem as $x\to 0$ and proved that that is actually true.