Since $f\left(x\,m x\right)\=\frac{m}{1\+m^2}\+x \sin \left(\frac{1}{m x}\right)$, the bivariate limit at the origin will be direction-dependent, and will therefore not exist.

The iterated limit $\underset{x\to 0}{lim}\left(\underset{y\to 0}{lim}f\right)$ also fails to exist: the inner limit fails to exist because of the infinite oscillations in $\sin \left(1\/y\right)$.

The iterated limit $\underset{y\to 0}{lim}\left(\underset{x\to 0}{lim}f\right)$ is necessarily zero because the inner limit is the limit as $x\to 0$ of $x$ times something that is finite, namely, $\frac{y}{x^2\+y^2}\+\sin \left(1\/y\right)$.