Every absolutely continuous function with integrable derivative tends to zero at infinity I am given $f,f' \in L^1(\mathbb{R})$, and f is absolute continuous, I want to show that:
$$\lim_{|x|\rightarrow \infty} f(x)= 0$$
Not sure how to show this, I know that $f(x)=\int_0^x f'(t) \, dt+f(0)$, and I can assume without loss of generality that $f(0)=0$, any help?
Thanks in advance.

Because $f'\in L^1$ you know that $\lim_{x\rightarrow \infty} f(x) $ exists (note $|\int_x^y f'| \le \int_x^y|f'| $ and $\int_x^\infty|f'|\rightarrow 0$ if $x\rightarrow\infty$). Since $f\in L^1$ the limit can only be zero, otherwise $\int|f|$ would not exist. 
(Of course the reasoning is the same for $x\rightarrow-\infty$)
