δ has to be a constant, it can't have the x inside it

oh I see, then how do I manipulate such that a constant is formed?

https://math.stackexchange.com/questions/1312410/proving-a-limit-of-a-constant-function

You can just let δ = a positive number which is ≤ ε . No matter which δ you choose here, ε always > 0. So |c - c| always < ε

The proof for the limit of a constant function is done differently from the usual

https://en.m.wikibooks.org/wiki/Calculus/Proofs_of_Some_Basic_Limit_Rules

It is that simple