How Do You Prove a Limit Using the ε-δ Definition?

[SithsNGiggles]

Homework Statement

Use the ε-δ definition of the limit to prove that

lim as x -> 3 of (2x - 6)/(x-3) = 2

Homework Equations

The Attempt at a Solution

I've started a preliminary analysis for the proof:
For any ε>0, find δ=δ(ε) such that 0 < |x - 3| < δ implies that |(2x - 6)/(x-3) - 2|< ε.
Simplifying:
|(2x - 6)/(x-3) - 2| = |(2x - 6 - 2(x - 3))/(x-3)| = |(2x - 6 - 2x + 6)/(x - 3)| = 0

Where do I go from here? Have I just proven that since the simplified form is 0, which is less than ε, and therefore proven the limit?

 

[SammyS]

What do you get when you reduce (2x-6)/(x-3) to lowest terms?

 

[SithsNGiggles]

2. Does that mean I've proved the limit? No formal proof required?

 

[SammyS]

No. Any δ > 0 will work, but you still need to show that for any x such that 0 < |x - 3| [STRIKE]< δ[/STRIKE] , |(2x - 6)/(x-3) - 2|< ε .

I crossed out the < δ side, because as long as (x-3) ≠ 0, (2x - 6)/(x-3) - 2 = 0 which, of course is less than ε .