Answer
$0$
Work Step by Step
Need to find the limit for $\lim\limits_{x \to \infty}
(e^{-2x}cosx)$.
$\lim\limits_{x \to \infty}(e^{-2x}cosx)=\lim\limits_{x \to \infty}\frac{cosx}{e^{2x}}$
The limit for $cosx$ lies between -1 to +1 any finite value.
Therefore, $\lim\limits_{x \to \infty}\frac{cosx}{e^{2x}}=\frac{\lim\limits_{x \to \infty}cosx}{\lim\limits_{x \to \infty}e^{2x}}$ since $-1\leq cosx\leq1$
Hence, $\lim\limits_{x \to \infty}(e^{-2x}cosx)$=finite value divided by $\infty$ approaches to $0$.