TITLE: How do you show that $\lim_{(x,y)\to(0,0)} \frac{xy}{x^2+y}$ doesn't exist?
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: Let $$m=\frac{xy}{x^2+y}$$
Then $$m(x^2+y)=xy$$ so $$y=\frac{mx^2}{x-m}$$
For any $m$ we have an equation of a curve, which meets $(0,0)$ and on which the expression $\frac{xy}{x^2+y}$ is constant, equal $m$. This way there are infinitely many paths leading to the origin, each of which results in a different limit $m$.
Hence the limit does not exist.