Geometrical meaning about a functuion of two variables

Question

Let $$F(x,y)=\frac{2+\cos(x-y)}{\sin(y-x)}$$

it's clear that the domain of definition of $F$ is $\mathbb{R}^2 -\{x=y\}$

What is the geometrical meaning of the right limit of $F$ following the line $ y=x+\pi$?

Answer 1

Note that the domain of definition is

$$\mathbb{R}^2 \setminus\{(x,y):y-x=k\pi\}$$

For the limit $t=y-x\to \pi^+$ we have

$$\frac{2+\cos(x-y)}{\sin(y-x)}=\frac{2+\cos(t)}{\sin(t)} \to\frac 1 {0^-}=-\infty$$