# How could I solve this limit with polar coordinates?

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.

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.
• For some bizarre reason people have a really hard time understanding this: if the limit exists, then computing it along every path will give the same result; so if one can find two paths which give different limits, then the original (full) limit does not exist. I gave you two paths which give different limits: either of the coordinate axes, and $x = t$ and $y = t^2/2$. So the original limit does not exist. – Ivo Terek Oct 31 '20 at 1:57