1549901369-Elements_of_Real_Analysis__Denlinger_

214 Chapter 4 • Limits of Functions

Similarly, we can combine finite and infinite limits algebraically. Suppose
P > 0 and N < 0 represent positive and negative real numbers, respectively,
which are limits of functions. Table 4.2 summarizes the results:

Table 4.2

Algebra of Infinite Limits

(+oo) + P(or N) = +oo

(-oo) + P(or N) = -oo

(±oo) · P = ±oo

(±oo) · N = =foo

( ±oo) · 0 is indeterminate

_1_ =0

±oo

1

The indeterminate

0

, covered by Theorem 4.4.3 along with Exercises 2 and

4, can be refined a bit further. Suppose limf(x) = 0. We shall write limf(x ) =
o+ if f (x) > 0 throughout appropriate interval( s), and lim f(x) = o-if f (x) < 0
throughout appropriate interval(s). With this understanding, we have

Table 4.3

Algebra of Infinite Limits

1

o+ = +oo

1

-=- 00

o-

In Section 4 .2 we proved the "Squeeze Principle" (Theorem 4.2.20) for

[finite] limits of functions. The analogous result for infinite limits is called the

"Comparison Principle,'' and is stated in the following theorem.

Theorem 4.4.10 (Comparison Test) Suppose that f(x) ~ g(x) for all x in
some deleted neighborhood of xo.

(a) If lim f(x) = + oo, then lim g(x) = + oo;

X--+Xo X--+Xo

(b) lim g(x) = -oo, then lim f(x) = -oo.

x--+xo x--+xo