Find $\lim\limits_{x\to -\infty} (e^{-x} \cos{x})$ $$\lim\limits_{x\to -\infty} (e^{-x} \cos{x})=\lim\limits_{x\to -\infty} \left(\dfrac{\cos{x}}{e^x}\right)$$
From there, I see that $e^x$ approaches $0$ while $\cos{x}$ oscillates between $-1$ and $1$.
My answer is that the limit does not exist. What is the proper reasoning to explain this? Does the limit oscillate forever, approach $\pm\infty$, etc.?

Take $x=\frac{\pi}{2}+\pi k$, where $k$ is integer and $k\rightarrow-\infty$.
We see that $e^{-x}\cos{x}=0.$
In another hand, for $x=\pi k$ and $k\rightarrow-\infty$ we have $|e^{-x}\cos{x}|\rightarrow+\infty$.
Id est, the limit does not exist.
