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Let $\displaystyle f(x,y) = \frac{x^2 y}{x^4 + y^2}$

Does this function have a limit when $(x,y) \to (0,0)$?

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See my (closely) related answer here. – Cameron Buie Mar 25 '13 at 9:51
Thank you. I worked it out and found that the limit dne but my tutor said that it did... – Charlie Brown Mar 25 '13 at 10:00
My answer there (and the comments below it) actually show you how to rigorously prove that the limit doesn't exist. Your tutor is mistaken. – Cameron Buie Mar 25 '13 at 10:04

Hint: Set first $x=\sqrt{y}$ for $y>0$, and then $x=y$.

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Hint $$\lim_{x\to0}f(x,x^2)=\frac{1}{2}\not=0=\lim_{x\to0}f(x,0)$$

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Let $x^2=r\cos\theta$ and $y=r\sin\theta$, where $-\pi/2\le \theta\le \pi/2$. Then our function is equal to $\cos\theta\sin\theta$. In particular, its values for $r$ near $0$ depend on the value of $\theta$.

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