Proving inexistence of limit Prove that limit does not exist
$$\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt{x^2+y^2}}$$
Obviously, since it is symmetric in $x$ and $y$, classic approach of substituing $x$ as a "simple" function - linear, polynomial, etc. Won't work. Polar gives us a reasonable answer 
$r\sin\theta\cos\theta$. How would I easily show it does not exist?

Using $$\frac{xy}{x^2+y^2}\leq \frac{1}{2}$$ we have that 
$$\left|\frac{xy}{\sqrt{x^2+y^2}}\right|\leq \frac{1}{\sqrt{2}}\sqrt{xy}\rightarrow 0$$
so the limit exists and is zero.
