TITLE: To find the limit of $\frac{1}{\sin n}+\frac{1}{\cos n}$
QUESTION [7 upvotes]: What should be the value of $\lim (\frac{1}{\cos n}+\frac{1}{\sin n})$ ? I think the limit does not exist.
Thanks in advance

REPLY [3 votes]: Let $x_n=1/\sin n+1/\cos n$. 

The jumps from $n$ to $n+1$ are of size $1$ and the size of a quadrant is $\pi/2\gt1$ so there are infinitely many integers $n$ such that $n$ mod $2\pi$ is in each of the four quadrants.
For every $n$ in the first quadrant $(0,\pi/2)$ mod $2\pi$, one knows that $0\lt\cos n\lt1$ and $0\lt\sin n\lt1$ hence $1/\cos n\gt1$ and $1/\sin n\gt1$ hence $x_n\gt2$.
For every $n$ in the third quadrant $(\pi,3\pi/2)$ mod $2\pi$, one knows that $-1\lt\cos n\lt0$ and $-1\lt\sin n\lt0$ hence $1/\cos n\lt-1$ and $1/\sin n\lt-1$ hence $x_n\lt-2$.

This proves that $(x_n)$ diverges.