Find $$ \lim_{n\to\infty}\int_\mathbb{R} nf(t)\cos(tn)e^{-n|t|}dt $$ assuming that $f:\mathbb{R}\to\mathbb{R}$ is continuous, bounded and integrable.
I know that before putting limit inside integral we need to do some substitution inside. But honestly, I do not know what substituion and why. So any hint would be gratly appreciated.
Maybe we could use substituion $t=u/n$, then $dt=du/n$ and $$ \lim_{n\to\infty}\int_\mathbb{R} nf(t)\cos(tn)e^{-n|t|}dt=2\lim_{n\to\infty}\int_0^{\infty} nf(t)\cos(tn)e^{-nt}dt=2\lim_{n\to\infty}\int_0^{\infty} f(u/n)\cos(u)e^{-u}du=2\int_0^{\infty}\lim_{n\to\infty} f(u/n)\cos(u)e^{-u}du=2\cdot f(0)\int_0^{\infty}\cos(u)e^{-u}du=f(0) $$ But I am not sure if it is what we are looking for.