设
$$
f(x, y)= \begin{cases}x \sin \frac{1}{y}+y \sin \frac{1}{x}, & x y \neq 0 \\ 0, & x y=0\end{cases}
$$
讨论下列极限：
(1) $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} f(x, y)$;
(2) $\lim _{x \rightarrow 0} \lim _{y \rightarrow 0} f(x, y)$;
(3) $\lim _{y \rightarrow 0} \lim _{x \rightarrow 0} f(x, y)$.