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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.

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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.

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  • $\begingroup$ 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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