Why doesn't the following limit exist?

$$\lim_{n\rightarrow \infty }\left(1+\frac{\cos(n\pi )}{n}\right)^{n}$$

My attempt:

Limit of $a_{2n}=(1+\frac{cos(2n\pi )}{2n})^{2n}$ is different from limit of $a_{2n+1}=(1+\frac{cos(2n\pi+\pi )}{2n+1})^{2n+1}$ because $\cos (n \pi)=(-1)^n$ which gives me $-1$ or $1$ and that's why limit doesn't exists, because cos oscilating between $-1$ and $1$

Is my answer correct?