epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

94
The Nuts and Bolts of Proof, Third Edition
Figure 2
-6-1
It is easy to check on the graph that when the value of x is in the interval
(1.75, 2.25) the value of/(x) will be closer than 0.75 to 1.
The definition of limit considered in this section assumes that both the
number c and the limit L are finite, real numbers. Appropriate definitions
can be stated to include dboo {i.e., to use the extended real number system)
and one-sided limits. This will not be done here, as it is beyond the goal of
this presentation. The main goal of this section is to provide the reader with
a first approach to the simpler examples of Hmits.
At this point, it might be useful to reread the definition of limit given
above and note some important facts.
1. The number ^>0 is assumed to be a given positive number, whose
value cannot be specified, as the statement must be true for all s>0.
This number can be very large or very small, and it provides the
starting point for "finding" the number 5 > 0.
2. The number 5>0, whose existence needs to be proved, will depend
on^ and usually on point c. In general, for a given s the choice of
the number 8 is not unique (see Exercises 4 and 5 at the end of this
section).
3. The function/might be undefined at c; that is, it might not be possible
to calculate/(c). Therefore, it is not always true that L =f(c).
4. The variable x can approach the value c on the real number fine from
its left (x < c) and from its right (x > c); no direction is specified or can
be chosen in the setting presented here.
EXAMPLE 2. Prove that lim
x^lX-
1
2-1
Proof: The goal is to prove that for every given positive number s there
exists a positive number 8 such that if x satisfies the inequahty \x— 1\<8,
then t((x-l)/(x2-l))-(1/2)1 <^.