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.

  • $\begingroup$ Ok. So the choice for x=t and y=(t^2)/2 is testing until it gives me a limit that proves otherwise, right? Thank you by the way! $\endgroup$ – Dottox Oct 31 '20 at 1:53
  • $\begingroup$ 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. $\endgroup$ – Ivo Terek Oct 31 '20 at 1:57

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