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

98 The Nuts and Bolts of Proof, Third Edition
Case 2. L>0. (We want to prove that the values of/(x) are "not very
close" to L. Because L is positive, we can try using negative values of/(x)
that correspond to negative values of x.)
-8/4
Let xs = -8/4, so that \X8 -0\<8. Note that f{xs) = i——r =
-S/4 ^ ^. . |-V4|
-—— = -1. Therefore,
8/4
\f(xs)-L\ = 1-1 - L| = |-(1 +L)| = 1 +L > 1 > f.
Thus, lim f{x) does not exist.
•
When proving that a limit does not exist, as done in Example 3, we need
to find a value of e so that, even for values of x close to c, \f(x) — L\ > S.
There is no "recipe" for doing so. In general, the value of £ that will enable us
to complete the proof depends on the behavior of the function around c, and
smaller values ofs are more likely to work. In Example 3, we chose s = 1/2,
but, if one looks carefully through the steps of the proof, any value of e
smaller than or equal to 1 would be acceptable. Thus, in general the choice of
£ is not unique.
The proof technique illustrated in Example 3 is not always easy to
implement. Pragmatically, as one advances in the study of real analysis, one
builds more and more tools to deal efficiently with the nonexistence of
limits. Some of these tools rely on the structure of the real numbers (e.g.,
density properties of rational and irrational numbers) and on the relationships
between functions and sequences. Therefore, while we will not examine
these topics in depth, we think it is useful to consider at least the definition
of limits of sequences. To use a self-contained approach, we will include
the definition of sequence as well.
Definition. A sequence of real numbers is a function defined from the
set of natural numbers N into the set of real numbers. The value of the
function that corresponds to the number n is the real number usually
indicated as a„. The number a„ is referred to as a term of the sequence.
Very often a sequence is identified with the ordered collection of real
numbers {«n}^i-
Definition. The real number L is said to be the limit of the sequence {a„}^i,
written as lim a„ = L, if for every 6>0 there exists a number Ng such that
|a„ — L\<s for all n>Ne.
The definition is stating in a formal way the idea that as the number
n gets "large enough," larger than a number Ng that depends on s, the
corresponding term an gets "closer than £" to the number L.