Im trying to calculate (or prove that the limit dosent exists)

$ \lim_{\left(x,y\right)\to\left(0,0\right)}\frac{xy^{2}}{x^{4}+y^{2}} $

Wolfram says the limit does not exists, but for every path I choose the limit ends up being 0.

If someone can tell if the limit exists or not, it would be helpful.

Thanks in advance

  • $\begingroup$ Try $y=x^t$ for some adequate value of $t$... or try to find this ADEQUATE value. $\endgroup$ – Tito Eliatron Oct 30 '20 at 18:29
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    $\begingroup$ $\left| \frac{xy^{2}}{x^{4}+y^{2}} \right| \le \left| \frac{xy^{2}}{y^{2}} \right| = |x|$, so ... $\endgroup$ – Martin R Oct 30 '20 at 18:31
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    $\begingroup$ See math.stackexchange.com/q/66226/42969 for a general result on this type of limit. $\endgroup$ – Martin R Oct 30 '20 at 18:34
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    $\begingroup$ You may have miscopied this. The usual would be $\frac{x^2 y}{x^4 + y^2} $ $\endgroup$ – Will Jagy Oct 30 '20 at 19:47
  • $\begingroup$ $$\left|\frac{xy^2}{x^4+y^2}\right|\le \frac{|x|(x^4+y^2)}{x^4+y^2}=|x|$$ $\endgroup$ – Mark Viola Oct 30 '20 at 20:03


By $x^2=u$ and $y=v$ we obtain

$$\left|\frac{xy^{2}}{x^{4}+y^{2}}\right|= \frac{\sqrt u v^{2}}{u^{2}+v^{2}} $$


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