Find limit of $\lim_{|n|\rightarrow\infty}\cos(in)$

Question

I need to find the limit of :
$\lim_{|n|\rightarrow\infty}cos(in)$ where $i=\sqrt{-1}$

What I did :
$\lim_{|n|\rightarrow\infty}\cos(in)=\lim_{|n|\rightarrow\infty}(e^{i^2n}+e^{-i^2n})/2=\lim_{|n|\rightarrow\infty}(e^n+e^{-n})/2=(\infty+0)/2=\infty$
But that doesn't seem to be true, because cosine must be between $-1$ and $1$. I used the fact that $\cos(x)=(e^{ix}+e^{-ix})/2$. What did I do wrong and how can I find the right answer ?

Thanks for your help !

Answer 1

Note that for $x\in \mathbb{R}$

$$\cos{ix}=\cosh x\to \infty$$

whereas, more in general, for $z\in \mathbb{C}$

$$\lim_{|z|\rightarrow\infty}cos(iz)$$

does not exist.