TITLE: How do I evaluate this limit :$\displaystyle \lim_{x\to \infty} (1+\cos x)^\frac{1}{\cos x}$?
QUESTION [6 upvotes]: I would like to know if this :$$ \lim_{x\to \infty} (1+\cos x)^\frac{1}{\cos x}$$ does exist and how do i evaluate it ?.
Note : I have tried to use the standard limit : $$ \lim_{z\to \infty} \left(1+\frac{1}{z}\right)^z=e$$ using $\cos x=1/z $ but i can't 
succeed 
Thank you for any help .

REPLY [6 votes]: For any natural number $n$, if $x=2n\pi$ ,the value  of the function is 2; if $x=(2n+1/2)\pi$, the value is 1. So no convergence.