1549901369-Elements_of_Real_Analysis__Denlinger_

2.4 Divergence to Infinity 87

We can improve our statement about the indeterminate ~ by introducing

a further bit of notation. Suppose {an} is a sequence such that an -. 0. If {an}

has a tail consisting of all positive numbers, then we write

If {an} has a tail consisting of all negative numbers, then we write

With this understanding, we have

Table 2.3

Algebra of Infinite Limits

1

(a) o+ = +oo

(b)

1

• = -(X)
  o-

FINAL CAUTION ABOUT ±oo:

Always remember that +oo and -oo are not real numbers. They
should not be expected to obey the rules that pertain to real numbers. They
are merely convenient symbols, which seem to obey some common algebraic
rules. They are intended for use only in connection with limits.

EXERCISE SET 2.4

1. In each of the following, a limit statement is given. In each case, answer
   the following questions:

(i) After how many terms are we guaranteed that Xn > 100 (or Xn <

-100)?

(ii) For arbitrary but unknown M > 0, after how many terms are we

guaranteed that Xn > M (or Xn < -M)?

n^2 +1

(a) lim fa= +oo (b) lim --= +oo

n->OO n->oo n + 1

1 - n

1

. 1 + 4n - n^3
(c) lim --= -oo (d) 1m = -oo
n->oo y'ri n->oo 3n