The problem I'm working on is the following:
$$\lim _{(x,y)→(0,0)} \frac{(x^2ye^y)}{(x^4+4y^2)}$$
I don't quite understand how to separate the equation to solve it. Because if I substitute directly by polar coordinates, it gives me zero, and the real answer is that the limit doesn't exist.
Along the coordinate axes we get zero, so this means that if we can find a path for which the limit along it is not zero, we may conclude that the limit does not exist.
So let $x = t$ and $y = t^2/2$. Then $$\lim_{t \to 0} \frac{t^2(t^2/2)e^{t^2/2}}{2t^4} = \lim_{t\to 0} \frac{e^{t^2/2}}{4} = \frac{1}{4}\neq 0.$$The choice of path was made to make the denominator as simple as possible, and it worked.