Learning:
The Journey of a Lifetime
A Cloud Chamber of the Mind

January 2006 Mathematics Notebook

An Example of a "Learning Process" Journal

1.2 Finding Limits Graphically and Numerically [p. 47 - 55]

Introduction: If the value of a function approaches a specific value as the independent variable approaches a value from above and below, then that specific value is called the limit of the function for that independent value.

Keywords: An Introduction to Limits, Limits that Fail to Exist, A Formal Definition of a Limit

History: none

Description: One should always try to approach a problem three ways: numerically (by constructing a table of values), graphically (by drawing a graph, either by hand or using technology), and analytically (using algebra or calculus).

"The existence or nonexistence of f(x) at x = c has no bearing on the existence of the limit of f(x) as x approaches c." [p. 48]

A limit of a function may fail to exist because of one of three conditions:

• the behavior differs from the right and left
• the behavior is unbounded
• the behavior oscillates

There are many interesting special functions in mathematics.

f(x) = 0, if x is rational

f(x) = 1 , if x is irrational

This is known as the Dirichlet function, and it has no limit for any real number c.

Augustin-Louis Cauchy (1789 - 1857) was the first person to assign a mathematically rigorous meaning to the idea of a limit. His epsilon-delta definition of limit is the standard used today.

pdf Files: Grabiner, J. V. (1983, March) Who gave you the epsilon? The American Mathematical Monthly.

Mathematica:

Summary: This section gives the epsilon-delta definition of a limit and focuses on the meaning of the definition. A limit for a specified function at a given point may or may not exist. A limit exists only if the function approaches a single finite number as the independent value approaches the given point.

Reflection: I still need to read the Grabiner article. Even following some of the examples, and certainly for some of the exercises, I feel a strong need to use pen and paper to go through some of the steps. Difficulties: I am feeling uneasy about 2 different situations. I am not on top of the examples for the epsilon-delta definition and I do not feel that I can just open Mathematica up and obain a graph or a limit of a function. This is a strong signal to stop until both situations are clarified.

Links: The following URL gives access to the pdf files referenced in the Larson text. Fantastic!

• http://www.maa.org/pubs/calc_articles.html

6:20 am Ballina NSW Australia

It has taken awhile to reorganize my files for 2006. I have made some notes for section 1.2 in the Larson calculus text, but I am not satisfied with my understanding of the examples and problems for the epsilon-delta formal definition of a limit. Secondly, I am not satisfied with my ability to quickly open mathematica and obtain a graph or a table for a given situation. I am going to take a short break and make myself a fresh cup of coffee and then return to this. I also want to have another look at the University of Illinois website for parametric plots. 8:15 am

9:35 am I want to have a look at Mathematica and refresh my mind about the commands for obtaining a plot of an equation. I also want to see how to obtain a table of values (possibly using a list command of some sort). 10:30 am

2:35 PM I completed the exercises for this section using Mathematica. I am feeling much better about my use of Mathematica at the moment. 3:05 PM

Total elapsed time for the day: 3 hr 20 min.