I am trying to find: $\lim_{(x,y)\to (0,0)}\frac{xy\sqrt{|xy|}}{x^2 + xy + y^2}$
I suspect that the limit does exist as the combined power of $x$ and $y$ is higher in the numerator than in the denominator, and I have noticed a pattern where this produces a limit, but the reverse case does not.
I have tried using polar coordinates $x = r\cos{\theta}, y = r\sin{\theta}$ and simplifying to get:
$f(r\cos{\theta}, r\sin{\theta}) = \frac{r(\cos{\theta}\sin{\theta})\sqrt{|\cos{\theta}\sin{\theta}|}}{\cos{\theta}\sin{\theta} + 1}$
I can't seem to get anywhere from here. I was trying to apply the squeezed theorem, but this expression seems like it needs to be simplified more before I can do that. Any hints on how to do that? Or am I barking up the wrong tree and need to try another approach?
Many thanks for any insights :)