Find the limit or show that it does not exist.

It doesn't exist.

$$\lim_{x\rightarrow \infty}(e^{-x}+2\cos{3x})=\lim_{x\rightarrow \infty}(e^{-x})+\lim_{x\rightarrow \infty}(2\cos{3x})=0+\lim_{x\rightarrow \infty}(2\cos{3x})$$

The limit of cos(3x) as x goes to infinity is undefined as cos(3x) continues to oscillate between maximum and minimum values of 1 and -1 forever.