Show that $\lim_{(x,y)\to (0,0)} {xy\over \sqrt{x^2+y^2}}$ exists

Question

Find the limit $$\lim_{(x,y)\to (0,0)} {xy\over \sqrt{x^2+y^2}}$$

By approaching the origin along both $x,y$-axis, I got the same result $0$.

So how can I prove the limit exists by epsilon-delta definition?

Answer 1

Note that we have $$\left|\frac{xy}{\sqrt{x^2+y^2}}\right|\lt\left|\frac{xy}{\sqrt{y^2}}\right|=|x|$$ hence the limit exists and equals zero.