1. ## find the limit

$\displaystyle \displaystyle \lim_{x\rightarrow -\infty} {e}^{4x} \sin x$
$\displaystyle \displaystyle \lim_{x\rightarrow -\infty} {e}^{4x} \sin x$
Use the fact that $\displaystyle -1 \leq \sin (x) \leq 1$ and consider the behaviour of $\displaystyle e^{4x}$.
let $\displaystyle a_n\to0$ as $\displaystyle n\to\infty,$ then if $\displaystyle b_n$ is bounded, we have $\displaystyle a_nb_n\to0$ as well.