I am trying to prove the limit I came up with:

$$\lim\_{x\to\infty}\left(\csc\dfrac{m}{n+x}-\csc\dfrac{m}{x}\right)=\dfrac{n}{m}$$

This fact came from the double generalization of the special case $\,m=\pi,\;n=1\,$ which has exceptional connections to the longest length of a $2n$-gon of radius $1$, $\csc\left(\frac{\pi}{x}\right)$ which allows you to prove that 
$$\lim\_{x\to\infty}\left(\csc\left(\frac{\pi}{x+1}\right)-\csc\left(\frac{\pi}{x}\right)\right)=\frac{1}{\pi}$$

When I was bored, I began graphing and came up with the above result after a bit. Thanks.