Multivariable limits with epsilon delta help?

Given ε > 0, we need to find ε > 0 such that
0 < √((x+1)² + (y-8)²) < δ ==> |xy - (-8)| < ε.

To this end, note that
|xy - (-8)|
= |((x+1) - 1)((y-8) + 8) + 8|, via 'adding 0' cunningly
= |((x+1)(y-8) + 8(x+1) - (y-8) - 8) + 8|, via FOIL
= |(x+1)(y-8) + 8(x+1) - (y-8)|
≤ |x+1| |y-8| + 8 |x+1| + |y-8|, via triangle inequality
= √(x+1)² √(y-8)² + 8√(x+1)² + √(y-8)²
≤ √((x+1)² + (y-8)²) √((x+1)² + (y-8)²) + 8√((x+1)² + (y-8)²) + √((x+1)² + (y-8)²)
< √((x+1)² + (y-8)²) + 8√((x+1)² + (y-8)²) + √((x+1)² + (y-8)²),
assuming that √((x+1)² + (y-8)²) < 1
= 10√((x+1)² + (y-8)²).

So given ε > 0, let δ = min{1, ε/10}. Then, 0 < √((x+1)² + (y-8)²) < δ
==> |xy - (-8)| < 10√((x+1)² + (y-8)²) < 10(ε/10) = ε.

I hope this helps!