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Find a limit or prove that it doesn't exists: $\lim\limits_{(x,y) \to(0,0)} \frac{xy^3-yx^3}{ x^4 + y^4}$

I tried to prove that it doesn't exist by finding two sequences with different limits, but I couldn't find them, so I suppose the limit exists, but unfortunately I don't have any idea how to prove it.

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  • $\begingroup$ Consider the limit along any path $(x, y) = (at, bt)$ where $t \to 0^+$. Can you show that this limit depends on $a$ and $b$? $\endgroup$
    – feralin
    May 8, 2017 at 19:39
  • $\begingroup$ Oh, i'm so dumb. I focused on limits like ($\frac {1}{n}$, $\frac {1}{n}^{2}$) etc. and didn't try examples like that... Thanks a lot! $\endgroup$
    – user444531
    May 8, 2017 at 19:46

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Consider the limit as $y=ax$, where $a$ is a constant. Then $$\lim_{x\to 0}\frac{x^4a^3-x^4a}{x^4+a^4x^4}=\frac{a^3-a}{1+a^4}. $$

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  • $\begingroup$ Oh, i'm so dumb. I focused on limits like ($\frac {1}{n}$, $\frac {1}{n}^{2}$) etc. and didn't try examples like that... Thanks a lot! $\endgroup$
    – user444531
    May 8, 2017 at 19:46

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