MthT 430 Chapter 5 Limits - An Equivalent Definition

This note is an attempt to clear up the confusion I (JL) probably created at the beginning of class October 10, 2007.

It is also closely related to problems 5.25 and 5.26 in Spivak.

In applying the definition of lim_{x → a} f(x) = L, it is sometimes convenient to multiply ϵ and/or δ by positive constants.

Limits - Definitions

Definition (Actual, p. 96). 

lim x → a  f(x) = L

means: For every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < δ, then |f(x) − L| < ϵ.

An Equivalent Definition of Limit

Definition 10⁶. 

lim x → a  f(x) = L

means: For every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < 10⁶δ, then |f(x) − L| < ϵ.

10⁶ ⇒ Actual: Fix ϵ > 0. By 10⁶, there is a δ > 0 such that, for all x, if 0 < |x −a| < 10⁶δ, then |f(x) − L| < ϵ. If 0 < |x −a| < δ, then 0 < |x −a| < 10⁶δ, so |f(x) − L| < ϵ.

Actual ⇒ 10⁶: Fix ϵ > 0. By Actual, there is an η > 0 such that, for all x, if 0 < |x −a| < η, then |f(x) − L| < ϵ. Let δ = 10⁻⁶η. Then δ > 0 iff η > 0. For all x, if 0 < |x − a| < 10⁶δ = η, |f(x) − L| < ϵ.

10⁶ ⇒ Actual: (another Proof) Fix ϵ > 0. By 10⁶, there is a ρ > 0 such that, for all x, if 0 < |x −a| < 10⁶ρ, then |f(x) − L| < ϵ. Let δ = 10⁶ρ. Then δ > 0 iff ρ > 0. For all x, if 0 < |x − a| < δ = 10⁶ ρ, |f(x) − L| < ϵ.

---

File translated from T_EX by T_TH, version 4.04.
On 22 Aug 2014, 14:24.