Why Read This Book? - Index of

220 Chapter 7 Functions of a Real Variable
before in our work with sequences when we defined limn→∞ an. However, our use
of the symbol was merely a formal one, because convergence was defined solely
in terms of the indexing set of positive integers. In this section, we want to take
the expression limx→a f(x) = L and replace either a or L (or both) with one of the
symbols ±∞. Up until this point, the fact that a and L are real numbers allowed
us to discuss ɛ-neighborhoods of L and δ-neighborhoods of a. With our current
definition of neighborhood, it makes no sense to talk about a neighborhood of
infinity, unless of course we decide to give this term meaning. We will do precisely
this. In fact, as we go, we will concoct extended, new meanings for old language
and revamp some of our visual imagery to show that there just might be some way
to understand infinity in a way that we can almost treat it as a number.
The language we will use in replacing a or L with ±∞ will go something like
this. In discussing limx→±∞ = L, we call these limits at positive or negative infinity.
When we discuss limx→a f(x) =±∞, we call these limits of positive or negative
infinity. Graphically, a limit at infinity corresponds to a horizontal asymptote in the
graph of f , and a limit of infinity corresponds to a vertical asymptote. There are
buckets and buckets of ways to combine and specialize the ideas we will discuss
here. With both +∞ and −∞, with either a or L or both being replaced by these
symbols, with two-sided and one-sided limits, we can define a whole bunch of new
terms. By hitting a few, you will catch on to what the others ought to be, so we will
not be exhaustive.
7.4.1 Limits at Infinity
Let’s begin by replacing a with +∞, because it almost exactly replicates the theory
of sequences. Since a sequence is just a real-valued function whose domain is the
positive integers, our way of graphing a sequence as we did in Figure 6.1 makes
convergence as n →∞easy to visualize as a horizontal asymptote of discrete
points in the plane. If we imagine filling in values of the function to other real
numbers so that the domain is some interval (a, +∞), then the following definition
seems to be a natural adaptation of Definition 6.2.1.
Definition 7.4.1 Suppose f is defined on some interval (a, +∞). Then we say
limx→+∞ f(x) = L provided for all ɛ>0, there exists a real number M such that
x>Mimplies |f(x) − L| 0, there
exists a threshold point in the domain beyond which all values of the function fall
in the ɛ-neighborhood of L. This definition gives rise to a whole slew of theorems
involving limits of functions at +∞, where the proofs are identical to those for
sequences.
Theorem 7.4.2 limx→+∞ c = c.
Theorem 7.4.3 limx→+∞ 1/x = 0.