Proof verification in multi-variable calculus using $\epsilon-\delta$ statement
Solution 1:
You must start from $\sqrt{(x-a)^2 + (y-b)^2} < \delta$. If it is, we have $|x-a| < \delta$ and $|y-b| < \delta$. Therefore, $$ |f(x,y)-L| < 2\delta + 3\delta = 5 \delta $$ Choose $\delta = \epsilon/5$.