Find a specific number δ>0 such that if x2 + y2 = δ2, then |x2+y2+3xy+180xy5 < 1/10 000.
Answer: Choose δ < 0.002
ε-δ def'n of limit: lim (x,y) → (a, b) f(x) = L if for every ε > 0 there exists a δ > 0 such that 0 < √(x-a)2+(y-b)2, |f(x) - L| < ε.
The Attempt at a Solution
So, f(x) = |x2+y2+3xy+180xy5 and because x2 + y2 = δ2, the limit is being taken as (x, y) → (0, 0) and L = 0.
I don't understand how I would start this problem because it's not rational, like I'm used to. I can do this:
(x) = |x2+y2+3xy+180xy5 = x2 + y2 + 3|x||y| + 180|x||y5|, but now I am stuck. Any help is greatly appreciated.