I need to find the limit by just using the $\epsilon$ and $\delta$ definition. The function is the following:
$$\lim_{(x,y)\rightarrow(2,-2)} 4x^2 -5y^2 $$
I already know that $\lim \rightarrow -4$ and that if I were to do the proof knowing the limit, it would be:
If $0 < \sqrt{(x-2)^2 +(y+2)^2} < \delta \Rightarrow |3x^2-4y^2+4|<\epsilon $
Can the limit be proved to exist even if I don't know the exact value? If so, how can I do it? Given the result of the limit, how can I manipulate the expression: $|3x^2-4y^2+4|$ so that I can give the $\epsilon$ and $\delta$ relationship? I read that I can use polar coordinates but I would like to know if I can do it without them.
Thanks for the help.