How to check if $e^{-2t}cos(2 \pi t)$ is periodic/non-periodic?

Question

So by graphing the function it is clearly non-periodic, but I would like to know how to solve it in a more mathematical way. Is there is a way to expand this function somehow that I forget?

Answer 1

The function has the value $1$ at $0$ and it approaches $0$ as $ t \to \infty$. So it cannot be periodic.

If it has period $p$ then $f(np)=f(0)=1$ for all $n$ so $f(np)$ does not tend to $0$.

Answer 2

If the function was periodic with period $T$, then the sequence $\left(e^{-2kT}\cos(2\pi kT)\right)_{k\in\mathbb Z_+}$ would be constant. Since$$\lim_{k\to\infty}e^{-2kT}\cos(2\pi kT)=0,$$we would then have$$(\forall k\in\mathbb Z_+):e^{-2kT}\cos(2\pi kT)=0.$$But $e^{-2\times0\times T}\cos(2\times\pi\times0\times T)=1$.

Answer 3

The function is continuous, but not bounded. Every continuous periodic function is bounded.