Course: Calculus I
Topic: Limits and Continuity
Subtopic: Delta-Epsilon Proofs


In this lesson we formalize the process of evaluating limits. Limits are a concept of "closeness". As the distance between x and c (distance delta) becomes small we are interested in whether or not the function output values get close (distance epsilon) to a specific limit. If we assume a limit does exist, delta-epsilon proofs allow us to mathematically show that it does (or does not). Performing a delta-epsilon proof has a very specific mathematical format that must be precisely followed. This method will be used to prove many theorems throughout the sequence of calculus courses.


By the end of this topic you should know and be prepared to be tested on:


Terms you should be able to define: delta, epsilon, implies, such that, subset, delta-epsilon proof statement (memorize it!)

Text Notes

The delta-epsilon proofs are extremely important to those of you who are mathematics majors and/or intend to take a year of Advanced Calculus after this 4-term sequence of Calculus. If that is your goal you should explore the interactive figures in your e-textbook in MML and try to understand the formal definition of limit. Take note of the formal definition of limit and be sure you understand the notation, symbols, and connection to the graphs. Try to follow the logic of the delta-epsilon proofs as they come up in the text throughout the rest of the term.

If higher maths is not your goal, you may SKIP the delta-epsilon proofs noting though that it may still benefit you to try to understand the connection between the graphs and the algebraic proofs.

In either case I will NOT include these proofs on a test.

Supplementary Resources (recommended)

My delta-epsilon proof template can be used as a template to write out the proof of a limit using the formal definition of limit. If you need a key to the mathematical symbols used in delta-epsilon proofs, see FAQs - What are the meaning of these math symbols? and the link to Greek letters listed in the following Q&A on that page.

Supplementary Resources (optional)

Video: The Definition of Limit, Cal State LA

Lesson: Formal Definition of Limit, Dale Hoffman's Contemporary Calculus

rev. 2020-09-21