2.2 Formal Treatment of Limits

In this lesson, you will learn about the epsilon-delta definition of limits and how to work with it.

Lesson Objectives

Upon completion of the lesson 2.2, you will be able to:

• • Recognize the symbolic (“epsilon-delta”) definitions of limits.

  • Find the value of delta > 0 for epsilon > 0, analytically.

  • Find the value of delta > 0 for a given positive number when positive infinity is the limit (or a given negative number when negative infinity is the limit).

  • Write a reasonably correct proof using the epsilon-delta definition of limit or the definition of an infinite limit.

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this module.

• 1. YouTube videos: Limit Definition – Epsilon Delta Part 1 (intervals and notation – a precursor to Part 2):

• 1. Part 2 (actual definition):

• 2. Part 3 (example):

• 1. YouTube videos: Epsilon Delta Limit definition

  2. Definition 1:

• 1. Definition 2:

Lesson Readings

The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module.

• 1. Delta-Epsilon Proofs [James Madison University]

  2. Epsilon-Delta Limit Proofs [UCLA] , by Prateek Puri (2010) (Overview and Epsilon-Delta proofs)

  3. Further Examples of Epsilon-Delta Proof [UC Berkeley]

  4. The Definition of a Limit [Paul's Online Math Notes] (Epsilon-Delta) and using it in Proofs (Paul’s Online Notes).

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

• 1. The Epsilon-Delta Definition of a Limit - Practice Problems (University of Houston)

  2. Drill – Verifying Limits (Visual Calculus)

1. Precise Limits of Functions as x Approaches a Constant

Additional Resources

Below are additional resources that help reinforce the content for this module.

1. Limits (formal definition) [MathIsFun]

2. How to Construct a Delta-Epsilon Proof [Milefoot]