Bridge to Abstract Mathematics: Mathematical Proof and Structures

4.3 THE LIMIT CONCEPT (OPTIONAL) 133

which pictures the vertical "6 band" about the line x = a, and the hori-
zontal "E band" about the line y = L, determined by these inequalities,
respectively.
Frequently in mathematics, a good way to get the meaning of definition-
to figure out what it says-is to consider what it doesn't say, and more
particularly, write out carefully its logical negation (this in addition to
knowing at least one example that satisfies the definition and one that does
not. At any level of mathematics, no matter how abstract, you should never
be satisfied with your level of understanding of a definition unless you are
familiar with at least one specific example that satisfies, and one that doesn't
satisfy, that definition, assuming, of course, that such examples exist.). This
is the case for the epsilon-delta definition of limit. Before proceeding, you
should review the general rule for negating multiply quantified predicates
(Theorem 3 and Example 5, Article 3.4) and attempt to write out the epsilon-
delta formulation of the statement L # lirn,,, f(x).

EXAMPLE 1 Characterize the statement L # lirn,,, f(x) in terms of epsi-
lon and delta.

Solution The rule for negating any quantified predicate is, "change each

V to 3, each 3 to V, and negate the predicate." In view of this rule, and

recalling from Chapter 2 the tautology -(p -* q) t, p A -4, we see that

L # lirn,,, f(x) is expressed as

There are two reasons for expecting that the characterization of L #

lirn,,, f (x), given in Example 1, might be easier to grasp than its nega-

tion (i.e., than the original epsilon-delta definition of limit). One is that

the string of quantifiers in this statement begins with the quantifier 3 rather

than V, thus lending greater concreteness when the definition is applied.

Second, the connective A replaces the connective -, in the negated defini-

tion; most readers probably concluded, in the course of Chapter 2, that A

is, on an intuitive basis, an easier connective to work with than +.

Let us now examine the geometric meaning of the negation of the

definition of limit in several examples. Throughout these examples, we will

pay particular attention to the relationship between the graph off, in the

"immediate vicinity" of x = a, and possibilities for a value of E that can be

chosen to prove that L # lirn,,, f (x).

2x + 1,

EXAMPLE 2 Consider the function f(x) = x20}. use an

x-1, x<O

epsilon-delta argument to prove that 1 # lim,,, f(x). hen-prove that

• 1 # lirn,,, f(x) and 0 # lirn,,, f(x). Finally, indicate how to gen-
  eralize these arguments to prove formally that lirn,,, f (x) does not exist.