<s>[INST] Limits of two variables functions I'm computing limits of two variables functions I do not manage to resolve them all.
4)$\lim_{(x,y) \to (0,0)} \dfrac{x^3-y^3}{x^2+2y^2} $
Here I use polar coordinates with $x=r\cos{\theta}$ and $y=r\sin{\theta}$ I finally get $\lim_{(r) \to (0)} \dfrac{rcos^3\theta - sin^3 \theta}{cos^2\theta +2sin^2 \theta} $ but I do not see how to continue.
6)$\lim_{(x,y) \to (0,0)} \dfrac{x^{1/3}y^2}{x^2+y^2+|x-y|} $
For 6 I began using polar coordinates but do not manage to determine it too, how would it be possible to compute them ?
Thank you 
 [/INST] Regarding 4) you have $x^2+2y^2 \ge x^2 + y^2$, $\vert x^3 \vert \le \vert x \vert (x^2+y^2)$ and $\vert y^3 \vert \le \vert y \vert (x^2+y^2)$. Consequently: $$\vert \frac{x^3-y^3}{x^2+2y^2} \vert \le \vert \frac{x^3-y^3}{x^2+y^2} \vert \le \frac{(\vert x \vert + \vert y \vert)(x^2+y^2)}{x^2+y^2} \le \vert x \vert + \vert y \vert$$ and $\lim\limits_{(x,y) \to (0,0)} \vert x \vert + \vert y \vert =0$ so $$\lim\limits_{(x,y) \to (0,0)} \frac{x^3-y^3}{x^2+2y^2} = 0.$$
5) is not well defined. How do you define the function when $x+y=1$? 
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