Do integrable functions vanish at infinity? If $f$ is a real-valued function that is integrable over $\mathbb{R}$, does it imply that
$$f(x) \to 0 \text{ as } |x| \to \infty?  $$
When I consider, for simplicity, positive function $f$ which is integrable, it seems to me that the finiteness of the "the area under the curve" over the whole line implies that $f$ must decay eventually. But is it true for general integrable functions?

Here is a beautiful counter-example: 
$$\int_{\mathbb R}\sin(x^2)\ dx=\sqrt{\frac\pi2}$$
Other more extreme examples
$$\int_{\mathbb R}x\sin(2^{|x|})\ dx$$
These rely on the Dirichlet test for convergence of a series/integral.
