Let $f$, a Lebesgue integrable function in $\mathbb{R}$ ($\int_{\mathbb{R}}|f| < \infty$). Let: $$ g(t) := \int_{-\infty}^{\infty} f(x)\sin(tx)dx $$ Show that $g$ is continuous (which I did), and that: $$ \lim_{|t| \rightarrow \infty} g(t) = 0 $$
Why is the second part correct?
Thanks!