Bridge to Abstract Mathematics: Mathematical Proof and Structures

4.3 THE LIMIT CONCEPT (OPTIONAL) 129

remove much of that difficulty. We will now "deliver" on that promise.

In this article, with the introduction to logic completed, we return to the

limit concept with three specific goals: (1) We review and attempt to put

into perspective a number of basic facts about limits (including the defini-

tion and basic properties of continuity) through a number of examples that

illustrate the three basic categories into which every limit problem falls.

(2) We attempt to help you to appreciate the geometric meaning of the

epsilon-delta definition. A major tool in this endeavor is a close analysis

of the logical negation of that definition. Principles of the predicate calcu-

lus, in turn, are indispefisable for formulating that logical negation. (3) We

attempt, by means of the material in this article, to pave the way for the

writing of epsilondelta proofs, a topic pursued further in Article 6.1.

REVIEW

In most elementary calculus classes the emphasis in the treatment of limits

is placed on a "working knowledge" of the concept; given a function y =

f(x) and a point x = a, it is hoped that students can learn to use intuition,

information from the graph of the function, and certain "rules of thumb"

to determine the value of lim,,, f (x). Seldom are students expected, at that

level, to bring the epsilondelta definition of limit explicitly into play in

solving a problem about limits. Before focusing on that definition, let us

review, first, the basic rules of thumb by which students are generally ex-

pected to handle limit problems. This, in turn, is best done by a description

of the three categories of answer to a limit problem, namely:

Type I: lim,,, f (x) exists and equals f (a).

Type II: lirn,,, f(x) exists but does not equal f(a).

Type Ill: lirn,,, f(x) does not exist.

Note that these categories are, in a logical sense, mutually exclusive and

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exhaustive; that is, every problem of the form %d lim,,, f(x)" falls into
exactly one of these three types.

Type 1. The easikst situation to deal with (and the situation most com-

monly seen initially in an elementary calculus class) in a limit problem is

u lim,,, f(x) exists and equals the value off at a" If a limit problem is in

this category, we solve it simply by plugging the specific value x = a into

the defining rule for f(x). If lim,,, f(x) = f (a), we say that f is continuous

at a The intuitive interpretation of "wntihuity at a" is that there is no

"hole" in the graph at a, nor is there a "break" in the graph in the "imme-

diate vicinity" of a. We will soon see that this intuitive description, although

useful, oversimplifies and can mislead.