Determine whether the function
$$
f(x,y)=\left\{\begin{array}{@{}l@{}}
(x^3-y^3)/(x^2+y^2) & \text{for }(x,y)\neq(0,0)\\
0 & \text{for }(x,y)=(0,0)\end{array}\right.
$$
is continuous.

To do so, I test whether the limit of the function $f(x,y)=\frac{x^3-y^3}{x^2+y^2}$ exists by plugging $y=mx$. If the limit gives the same value for all $m$, it means that the function exists and hence is continuous. I'm unsure whether my method is correct.