$f, f'\in L^{1}(\mathbb R) \implies \lim_{x\to \infty} f(x)=0 ?$ Suppose $f\in L^{1}(\mathbb R) $ and $f'\in L^{1}(\mathbb R).$
My Question is: 

Can we show, $\lim_{x\to \infty} f(x)=0$ ?

Thanks,

Yes. Two steps:
Step 1. using the inequality
$$y>x>a\Longrightarrow  \vert f(x)-f(y) \vert\leq\int_a^\infty\vert f'(t)\vert dt$$
we conclude that $f$ satisfies Cauchy criterion as $x\to\infty$ (because $f'\in L^1$,) and consequently the limit $\lim\limits_{x\to\infty}f(x)=\ell$ does exist.
Step 2. the limit $\ell$ must be zero because otherwise $f$ would not belong to $L^1$, and this is an easy task.
