Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

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

Overview

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.

Objectives

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

• 1.4.1 Understand the symbols in and meaning of the delta-epsilon proof statement
• 1.4.2 Understand the graphical significance of the delta-epsilon proof statement
• 1.4.3 Perform delta-epsilon proofs on limits involving functions that are linear, quadratic, rational, radical, exponential, trigonometric, etc.
• 1.4.4 Use delta-epsilon proofs to show that a limit does or does not exist

Terminology

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)

rev. 2020-09-21