Introduction to Limits (ε, δ definition) (Calculus) (Multivariable Calculus #9) #4.3.2.2b1

This is the first of a short series on limits in one variable. In this video we start with an intuitive definition of the limit, then introduction ε, δ definition, then examine one-sided limits. Next time we'll look at the limit laws, then limits involving infinity, then continuity. Then we will return to multivariable calculus.

For the next several weeks we will be developing multivariable calculus, so we can acquire Green's Theorem, which is required for complex integration. Complex analysis is central to the proof of Fermat's Last Theorem.

With the first videos on this channel, we managed to keep the math very simple, almost exclusively nothing more advanced than algebra. Now, to complete our brief tour of elliptic curves, we have to take an ENORMOUS leap forward in the level of the mathematics involved.

I want to make sure we are all on the same page before we begin learning the more advanced stuff in earnest. I am going to assume we are all familiar with high school math and college math through first or second year calculus. I'm going to introduce complex numbers and complex integration in some detail with the next several videos, after examining multivariable calculus.

This series will culminate with the Weierstrass Equation, which is the thing that connects elliptic curves to complex numbers, and thus allows us to connect them to modular forms, which is what Wiles's proof is all about.

Here is the outline of this series on complex numbers:
0. Introduction
1. The Real Number System
2. Multivariable Calculus (we are here)
3. Complex Numbers
4. Complex Functions
5. Exponential and Trigonometric Functions
6. Complex Integration
7. Cauchy's Integral Theorem
8. Cauchy's Integral Formula
9. Laurent Series
10. Complex Residues
11. Lattices and Doubly Periodic Functions
12. Lattices and Tori and Groups
13. The Weierstrass p-Function
14. The Weierstrass Equation: Complex Functions and Elliptic Curves

0:00 - Introduction
0:36 - Intro to Limits
2:14 - Intuitive Definition
13:03 - Precise Definition (ε, δ)
28:32 - One-Sided Limits

Please leave any questions, comments, or suggestions in the comments below!

Credits:
Music: "Home" and "St. Jarna" by Depeche Mode

Bibliography:
""Calculus, Early Transcendentals," by James Stewart

[billhalprin]

Epsilon is the rise and delta is the run. If for any f(x), is there any reason not to compute the relation between epsilon and delta as the relation of the rise to the run?