Prove: using $\epsilon$-$\delta$ definition, the limit of both $f$ and $g$ as $(x,y)\to (0,0)$ is $0$.
$f(x,y)=xy$
$g(x,y)=\frac{xy}{x^2 +y^2+1}$
Also, for Q2 can I convert $g(x,y)$ to $m(x,y)/n(x,y)=g(x,y)$ using arithmetic of limits, then prove using $\epsilon$-$\delta$ definition the limit of function $m$ and $n$ separately; then combine the two?
Thanks :)
I wonder if this is correct: $|xy-0|<\epsilon$ given $|x-0|< \delta $ and $|y-0|< \delta $
$|xy-0|< |x-0||y-0|<\delta^2=\epsilon$
therefore: $\delta<\epsilon^{1/2}$