Epsilon-Delta continuity proof

[gomes.]

http://img34.imageshack.us/img34/1989/analysis123523456.jpg


I'm trying to work through some examples, but I am not sure where the following comes from:



1. circled in black -- how do i get the δ<1?



2. circled in red -- how do I get 0<x<2, i.e. x∈(0,2)?



3. cirlced in blue -- how do i get |x^2+x+3|<9



Thanks, most appreciated.

 

[I like Serena]

Let's start with 1 and 2.

delta is defined to be between 0 and min(1, epsilon/9).
min(x, y) means the minimum of x and y.
This means that delta < 1 and perhaps a lot smaller.

With delta < 1 and |x-1| < delta, what values can x take?