# Thread: why does this limit not exist?

1. ## why does this limit not exist?

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 $1$

2. ## Re: why does this limit not exist?

$\lim \limits_{n \to \infty} \left(1 + \dfrac{x}{n}\right)^n = e^x$

If the limit did exist

$\lim \limits_{n \to \infty} \left(1 + \dfrac{\cos(n\pi)}{n}\right)^n = e^{\cos(n\pi)}$

and as you alluded to this will forever oscillate between $e$ and $e^{-1}$ so no limit exists.