Question about the delta-epsilon definition of a limit

[swampwiz]

The definition says:

A finite limit exists if for every ε, there exists a δ such that if

a - δ < x < a + δ

then

| f( x ) - L | < ε

It seems that an equivalent statement would be:

A finite limit exists if for every δ that defines a domain region

a - δ < x < a + δ

that the function value is limited to a range region

| f( x ) - L | < ε

Thus it could also be said that for any domain region width δ (i.e., in the "neighborhood"), then if the function value is limited to some range region L ± ε, then the finite limit exists

Would this last statement be proper?

there exists an ε such that

 

[Citan Uzuki]

No, these are not the same thing. The definition of a limit is that "For every ε, there exists a δ such that a-δ < x < a+δ ⇒ |f(x)-L|<ε" Your statements both amount to saying "For every δ, there exists an ε such that a-δ < x < a+δ ⇒ |f(x)-L|<ε". Notice the change in word order? That matters. The point of the limit definition is not that the function will be contained in some small interval around L, but rather that you can force it to be contained in any interval around L by taking δ small enough. That's why the universal quantifier is in front of the epsilon, rather than the delta.