Answer
$0$
Work Step by Step
$-\dfrac {1}{e^{3t}}\leq \dfrac {\cos t}{e^{3t}}\leq \dfrac {1}{e^{3t}}$
$\Rightarrow \lim _{t\rightarrow \infty }-\dfrac {1}{e^{3t}}=\dfrac {1}{e^{3t}}=0\Rightarrow \lim _{t\rightarrow \infty }\dfrac {\cos t}{e^{3t}}=0$