解答题 （共 6 题 ），解答过程应写出必要的文字说明、证明过程或演算步骤
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) $\lim _{y \rightarrow 0} \lim _{x \rightarrow 0} f(x, y)$.
求极限$\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{\left(x^2+y^2\right) \sin \left(x y^2\right)}{1-\cos \left(x^2+y^2\right)}$
求极限$\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \ln (1+x y)^{\frac{1}{x+y}}$
求极限$\lim _{\substack{x \rightarrow+\infty \\ y \rightarrow+\infty}}\left(\frac{x y}{x^2+y^2}\right)^{x^2} \sin (x y)$
求极限: $\lim _{\substack{x \rightarrow \infty \\ y \rightarrow \infty}} \frac{x+y}{x^2-x y+y^2}$
求极限 $\lim _{(x, y) \rightarrow(0,0)}\left(\frac{x y}{\sqrt[3]{x^3+y^3}}+\frac{x^5}{y-x}\right)$