Question

What is limit of (x,y) tends to (0,0) 1/sinxy?

Answer 1

Along $x=y$

$$\lim\limits_{(x, y)\to(0,0)}(\dfrac{1}{\sin(xy)})=\lim\limits_{(x, y)\to(0,0)}(\dfrac{1}{\sin(x^2)})=\infin$$

Along $x=-y$

$$\lim\limits_{(x, y)\to(0,0)}(\dfrac{1}{\sin(xy)})=\lim\limits_{(x, y)\to(0,0)}(\dfrac{1}{\sin(-x^2)})=-\infin$$

Therefore $\lim\limits_{(x, y)\to(0,0)}(\dfrac{1}{\sin(xy)})$ does not exist.