Limit Proofs: Understanding Epsilon-Delta Proofs for Calculus

[new_at_math]

I just have a general question about limit proof using epsilon delta proofs

It generally follows the algorithm from this website:http://www.milefoot.com/math/calculus/limits/DeltaEpsilonProofs03.htm

focusing on the first example(example using linear functions)
The first table I get, but is the second table necessary, it basically restates the given?
it seems analogous to x + 1 = 3
x = 2
and then subbing 2 in for x

 

[CompuChip]

The first table is basically your working version that you scribble on the back of an envelope to find the value of $\delta$.
The second table is the actual proof that you can publish (or write on an exam). Note that it is basically your "working version" backwards, i.e. whereas you derived delta to make sure the proof will work, in the formal proof you just state it as if it appeared by magic and then proceed to show that it actually works.