1. ## Epsilon-Delta continuity proof

http://img34.imageshack.us/img34/198...s123523456.jpg
I'm trying to work through some examples, but im 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.

2. Originally Posted by rlkmg
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
1. In these proof, we are free to pick $\displaystyle \delta.$

2. If $\displaystyle |x-y|<\delta$ then $\displaystyle y-\delta<x<y+\delta$ so $\displaystyle |x-1|<\delta<1$ means that $\displaystyle 0<x<2$.

3. If $\displaystyle x\in (0,2)$ then $\displaystyle |x^2+x+3|<4+2+3$

3. Thanks. The only thing im wondering is..how do I know how to choose δ? what if i choose δ>1 etc.? How do i know what to choose?

Thanks

4. Originally Posted by rlkmg
Thanks. The only thing im wondering is..how do I know how to choose δ? what if i choose δ>1 etc.? How do i know what to choose?
Why do that?
You want the proof to work. Choice depends on what works.

5. sorry, i dont really understand what that part means.

Lets say i choose:
δ<2, and |x-1|<2
then -1<x<3

instead of 0<x<2 as done in the question.

So how would i know how to choose δ<1 instead of say δ<2? etc..