<s>[INST] how to show limit of $\frac{x^2-y^2}{x^3-y^3}$ as (x,y) goes to $(0,0)$ does not exist I know that $\lim\limits_{(x,y)\to (0,0)} \frac{x^2-y^2}{x^3-y^3}$ does not exist, but i'm not sure how to show it.
What I have done is let $y=kx$
and then,
$\lim\limits_{x\to 0} \frac{x^2-(k x)^2}{x^3-(kx)^3} = \lim\limits_{x\to 0} \frac{1-k^2}{x(1-k^3)}$
and k is just a constant,
is this enough to show that the limit does not exist?
 [/INST] Using polar coordinates, $x=r\cos\theta, y=r\sin\theta,$ we have $$\frac{r^2(\cos^2\theta - \sin^2\theta)}{r^3(\cos^3\theta-\sin^3\theta)}=\frac{\cos\theta + \sin\theta}{r(\cos^2\theta+\cos\theta\sin\theta+\sin^2\theta)}=\frac{\cos\theta+\sin\theta}{r(1+\cos\theta\sin\theta)}$$
Choose any $\theta$ such that the numerator is not $0$. Then the limit is undefined since the denominator tends to $0$ as $r\to 0.$
On the other hand, if you choose $\theta$ such that the numerator is zero, then the entire limit is $0$.
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