Suppose $f\in L^p[0,\infty)$ for some $p>0$. Also suppose that $f$ is absolutely continuous and $f'\in L^1$.
Show that $\lim\_{x\to\infty}f(x)=0$.
Since $f$ is absolutely continuous thus we have $|f(x)-f(0)|\le\int^x\_0 |f'(t)| dt\le||f'||\_{L^1}$. Thus we can only show that $f$ is bounded. How does it vanish at infinity?