When there are some conditions on the power of a rational function, how can we evaluate/prove the non-existence of the limit of the function? Here is a specific example :
Prove $$\lim_{(x,y)\to(0,0)} \frac{(x-y)^{p-1}((p-2)x^2+2xy+py^2)}{(x^2+y^2)^2}=0$$ if $p>3$.
Is this even solvable?
Hint: The expression is bounded above by
$$ \frac{(|x|+|y|)^{p-1}((p-2)|x|^2+2|x||y|+py^2)}{(x^2+y^2)^2}.$$
Let $r=(x^2+y^2)^{1/2}.$ Use $|x|,|y|\le r$ to finish.