Answer
The limit is
$$\lim_{(x,y)\to(2\pi,4)}\sin\frac{x}{y}=1.$$
The function is discontinuous at $y=0$ i.e. the $x$ axis.
Work Step by Step
We can find the limit by the simple substitution
$$\lim_{(x,y)\to(2\pi,4)}\sin\frac{x}{y}=\sin\frac{2\pi}{4}=\sin\frac{\pi}{2}=1.$$
This function is discontinuous at the line $y=0$ (this line is the $x$ axis) because we would have zero at the denominator and since we cannot divide by zero this gives
$$\lim_{(x,y)\to(x_0,y_0)}\sin\frac{x}{y}\neq\sin\frac{x_0}{y_0},$$
when $y_0=0$.
At all other points (where $y_0\neq0$) we can always substitute and get
$$\lim_{(x,y)\to(x_0,y_0)}\sin\frac{x}{y}=\sin\frac{x_0}{y_0}$$
and there the function is continuous.
