Why Read This Book? - Index of

7.4 Limits Involving Infinity 221
Theorem 7.4.4 Suppose f(x) → L1 and g(x) → L2 as x →+∞. Then the
following hold as x →+∞.
1. f(x) + g(x) → L1 + L2
2. f(x)g(x) → L1L2
3. f(x)/g(x) → L1/L2 (if L2 �= 0)
On and on the theorems go that exactly parallel our previous work. The limit
of polynomial over polynomial, and even a sandwich theorem, seem strangely
translucent.
EXERCISE 7.4.5 Create a definition for limx→−∞ f(x) = L for a function
defined on an appropriate interval.
Now let’s extend our language of neighborhood to include infinity. When we
say f(x) → L as x → a, we mean that any neighborhood of L has a corresponding
deleted neighborhood of a that maps into it. Is there a way to use the same
language for a limit at infinity? Could we say that any neighborhood of L has a
corresponding neighborhood of +∞ that maps into it? We can of course, if we
define a neighborhood of +∞ to be an interval of the form (M, +∞). Similarly, a
neighborhood of −∞ could be defined as an interval of the form (−∞,M).
We are catching a glimpse of the extended real numbers and are developing an
imagery of two phantom points ±∞ somewhere way off the left and right ends of
the number line. This is the standard way of creating the extended real numbers,
which are denoted as either R or [−∞, +∞].
An apparent difference between neighborhoods of a real number and neighborhoods
of ±∞ is that the latter are not two-sided. However, if we use the single
symbol ∞ instead of both ±∞, then we can create a nice way of visualizing the
extended real numbers, where +∞ and −∞ are merged into one point. Here is
one way we might do that, which is similar to the extension of the xy-plane to
include the point at infinity in complex analysis.
Imagine a real number line with a circle sitting on top of it as in Figure 7.5.
We map a given real number a to a corresponding point (x, y) on the circle with
the help of the diagonal line in the figure. This geometric way of mapping each
1
1
(x, y)
1 a
Figure 7.5 Mapping the extended real numbers onto a circle.