What is $\lim_{(x,y)\to(0,0)} \frac{(x^3+y^3)}{(x^2-y^2)}$? In class, we were simply given that this limit is undefined since along the paths $y=\pm x$, the function is undefined.
Am I right to think that this should be the case for any function, where the denominator is $x^2-y^2$, regardless of what the numerator is?
Just wanted to see if this is a quick way to identify limits of this form.
Thanks for the discussion and help!

For your function, in the domain of $f$ (so $x\ne \pm y)$, to compute the limit you can set $x=r\cos\theta, y=r\sin\theta$, and plug it it. You get $\lim\limits_{r\to 0} \frac{r^3(cos^3\theta+sin^3\theta)}{r^2(cos^2\theta-sin^2\theta)} =\lim\limits_{r\to 0} \frac{r(cos^3\theta+sin^3\theta)}{(cos^2\theta-sin^2\theta)}$, and you can easily see that this is $0$ for any $\theta$ in the domain of $f$ (you need to avoid $\theta = \frac{n\pi}{2}-\pi/4$). Of course, if you are considering the whole plane, then the limit does not exist, because the function isn't even defined at $y=x$, so you can't compute the limit along that path.
