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

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.

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

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