Evaluate the limit along the parabolas $y\=m x^2$, that is, evaluate

Since the limit depends on the direction of approach to the origin, the bivariate limit at the origin does not exist.

The "inner" limits for both iterated limits are zero, so the "outer" limits are taken of zero, with the result that both the iterated limits are zero.

Since the limit depends on the direction of approach to the origin, the bivariate limit at the origin does not exist. Alternatively, access Maple's bivariate limit through the Context Panel.

Maple's declaration that the limit is undefined is equivalent to the more prevalent statement that the limit does not exist.

It is possible to gain some insight into the near-origin behavior of this function by changing to polar coordinates.

$f\left(r \cos \left(\mathrm{\theta }\right)\,r \sin \left(\mathrm{\theta }\right)\right)\=\frac{\cos \left(\mathrm{\theta }\right)\sin \left(\mathrm{\theta }\right)}{{\cos \left(\mathrm{\theta }\right)}^{3}r\+{\sin \left(\mathrm{\theta }\right)}^2}$

which tends to $\cot \left(\mathrm{\θ}\right)$ as $r\to 0$. The unbounded behavior of the cotangent in the interval $\left[0\,2 \mathrm{\π}\right]$ should help explain why the bivariate limit does not exist at the origin.