Formal approach to limit

A real number l is limit of f(x) as x tends to a, if for every ε>0 there exists a δ>0 such that

0<|x-a|< δ => |f(x)-l|< ε
<=> x.( a- δ, a+δ), x≠a => f(x).(l- ε, l+ ε)

| f(x)-l| < ε means f(x) belong to the ε neighbourhood of l and | x-a| < δ (and x≠a) means x belongs to the deleted δ neighbourhood of a.

The open interval (a- δ, a+ δ) whose length is 2 δ and whose midpoint is a, is called the δ-interval of a. Open interval means a- δ, and a+ δ are not part of the interval.

{x/x Є(a- δ,a+ δ), x≠a} is called the deleted δ-neighbourhood of a.