Epsilon delta proof of a hard fraction.

Question

$\lim_{(x,y)\to (0,0)} \frac{xy}{x+y} $

Find the limit and prove using $\epsilon /\delta $ proof that it is the limit. i managed to factor out some stuff and show it converged to a limit value i was also reasonable sure the whole limit was zero so this limit should be zero but i can't prove it using $\epsilon /\delta $ proof.

Any help/hints much appreciated.

Answer 1

Let $y=x^2-x$, so that $x\to0$ ensures $(x,y)\to(0,0)$.

Then $$\frac{x(x^2-x)}{x+x^2-x}=x-1\to-1,$$ which contradicts the limit $0$.

Answer 2

Note that

for $x=t$ and $y=t^2-t$ for $t\to0$

$$\frac{xy}{x+y}=\frac{t^3-t^2}{t^2}=t-1\to -1$$

for $x=0$

$$\frac{xy}{x+y}=0$$

thus the limit does not exist.