Does $\lim_{(x,y)\to(0,0)}[x\sin (1/y)+y\sin (1/x)]$ exist?

For $x\ne0$ and $y\ne0$ we have $$ \left|x\sin\frac{1}{y}+y\sin\frac{1}{x}\right|\le \left|x\sin\frac{1}{y}\right|+\left|y\sin\frac{1}{x}\right|\le|x|+|y| $$ So, for $$ f(x,y)=\begin{cases} x\sin\dfrac{1}{y}+y\sin\dfrac{1}{x} & \text{if $x\ne0$ and $y\ne0$} \\ 0 & \text{if $x=0$ or $y=0$} \end{cases} $$ we have $$ |f(x,y)|\le |x|+|y| $$ for all $(x,y)$. Therefore $$ \lim_{(x,y)\to(0,0)}f(x,y)=0 $$ by the squeeze theorem.

Be careful that $x\sin(1/y)\le x$ is not true in general, but you just need the absolute value and $|x\sin(1/y)|\le|x|$ is true (provided $y\ne0$, of course).

WolframAlpha is a great resource, but it doesn't always tell the truth. ;-)