# Find $\lim\limits_{(x,y) \to(0,0)} \frac{xy^3-yx^3}{ x^4 + y^4}$

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.

• Consider the limit along any path $(x, y) = (at, bt)$ where $t \to 0^+$. Can you show that this limit depends on $a$ and $b$? – feralin May 8 '17 at 19:39
• 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! – user444531 May 8 '17 at 19:46

Consider the limit as $y=ax$, where $a$ is a constant. Then $$\lim_{x\to 0}\frac{x^4a^3-x^4a}{x^4+a^4x^4}=\frac{a^3-a}{1+a^4}.$$
• 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! – user444531 May 8 '17 at 19:46