1549901369-Elements_of_Real_Analysis__Denlinger_

4.4 *Infinity in Limits 213

(e) Suppose lim f(x) = +oo and lim h(x) = -oo. Let M > 0. Since

x-~ x-~

lim f(x) = +oo, 3 81 > 0 3 0 < Ix - xol < 81 =? f(x) > M. Since lim h(x) =

X--+Xo X--+Xo

-oo, 302 > 0 3 0 <Ix - xol < 82 =? h(x) < -1. Leto= min{o 1 , 82 }. Then,

0 <Ix - xol < o =? f(x) >Mand h(x) < -1

=? f(x) >Mand - h(x) > 1

=? f(x) (-h(x)) > M · 1

=? - (f(x)h(x)) > M

=? f(x)h(x) < -M.

Therefore, by Definition 4.4.1, lim (f(x)h(x)) = -oo. •

X--+Xo

Corollary 4.4.9 Theorem 4.4.8 remains true when x ---+ x 0 is replaced by

x ---+ x- +

0 or x ---+ x 0.

Symbolic Shorthand: The results of Theorem 4.4.8 and its corollary are
often expressed as a kind of "algebra" of +oo and -oo, summarized in Table
4.1 as follows:

Table 4.1

Algebra of Infinite Limits

(+oo) + (+oo) = +oo

(-oo) + (-oo) = -oo

(+oo) · (+oo) = +oo

(-oo) · (-oo) = +oo

(+oo) · (-oo) = -oo

Caution: In Table 4.1, the symbols +oo and -oo are not to be regarded
as numbers. They cannot be manipulated as numbers, nor can they be expected
to obey the usual rules of algebra. They represent limits only.

Indeterminate Forms: The forms ( +oo) + ( -oo) and ( +oo) - ( + oo)
are "indeterminate" in the sense that no answer can be given that is always
true. That is, there are pairs of functions, f(x) and g(x), such that limf(x) =
+oo and limg(x) = -oo for which lim[f(x) + g(x)] = +oo, others for which
lim[f(x) + g(x)] = -oo, others for which lim[f(x) + g(x)] is a finite number,
and still others for which lim[f(x) + g(x)] does not exist.