Thomas’ Calculus Early Transcendentals 13ed

Supplement to Lecture 02

The precise definition of lim_{x→c} f(x) = L is

For any ε > 0, there exists a (corresponding) δ > 0 such that,

0 < |x − c| < δ ⟹ |f(x) − L| < ε.  (1)

In practice, however, we only need to show that (1) holds for small enough ε > 0. This is convenient in many examples. See for instance, the square root in Example 5 of section 2.3.

In other words, we can use

For any ε ∈ (0, ε_{0}), ε_{0} > 0, there exists a (corresponding) δ > 0 such that,

0 < |x − c| < δ ⟹ |f(x) − L| < ε.  (1)

The reason is, suppose that (1) holds for all ε ∈ (0, ε_{0}), where ε_{0} > 0. Then in particular, it holds for ε = ε_{0}/2 and therefore we can find the corresponding δ in (1). Denote this particular δ by δ_{0}, then the statement (1) reads,

0 < |x − c| < δ_{0} ⟹ |f(x) − L| < ε_{0}/2.  (2)

This implies that (1) not only holds for ε ∈ (0, ε_{0}), it actually holds for any ε > 0. To see this, we simply take δ = δ_{0} for those ε ≥ ε_{0}. Then from (2), we have

0 < |x − c| < δ_{0} ⟹ |f(x) − L| < ε_{0}/2 < ε.  (3)

This shows that taking δ = δ_{0} works for those ε ≥ ε_{0}.