Q:

Does the following limit exist $\lim_{(x,y)\rightarrow (0,0)}\frac{\sin(xy)}{y}$

Does the following limit exist? 

$\lim_{(x,y)\rightarrow (0,0)}\frac{\sin(xy)}{y}$

The answer here is not correct in my opinion since it does not take under considerarion all the surface parts in which $y=0.$

A:

Set $g(x)=\begin{cases} \frac{x\cdot \sin(xy)}{xy} \quad \text{if}\quad x\neq 0\\ 0 \quad \quad \quad \;\; \text{if}\quad x=0 \end{cases}$
Then $$\lim_{(x,y)\rightarrow (0,0)}\frac{\sin(xy)}{y}=\lim_{(x,y)\rightarrow (0,0)}g(x)=0,$$ as we can make $\frac{\sin(xy)}{xy}$ as close to $1$ as we wish. 

