I have to prove that this limit doesn't exist.
$$\lim_{(x,y)\to(0,0)} \frac{xy}{x^2+y}$$
I tried this parametrization: $\begin{cases} x = t \\ y = mt^\alpha\end{cases}$
obtaining as result that the previous limit in this specific case would be equivalent to
$$\lim_{t\to0} \frac{mt}{t^{2-\alpha}+m}$$
which would be null for each value of $\alpha,m$.
Using a polar coordinate system doesn't seem effective too.
How do I prove that this doesn't exist?
Best Answer
Let $y=-x^2+x^4$. Then,
$$\frac{xy}{x^2+y}=-x^{-1}+x$$
What happens now?