1549901369-Elements_of_Real_Analysis__Denlinger_

220 Chapter 4 • Limits of Functions

(d) Suppose f(x) < 0 for all x in some neighborhood of -oo. Then

1

x~-~ lim f(x ) = -oo <=? x~-~ lim f( X ) = 0.

Proof. Exercise 6. •

. 1
Example 4.4.22 Prove that \fn EN, hm - = 0.
X->± 00 Xn

Proof. In Theorem 4.4. 18 (a) we proved that lim xn = +oo. Thus,

x->+oo

lim ~ = 0 by Theorem 4.4.21 (a). The proof of lim ~ = 0 requires two

x->+oo xn x->- 00 xn

cases: when n is even, and when n is odd. Use Theorem 4.4.18 (b), (c), and

Theorem 4.4.21 (c), (d). (Exercise 7.) D

ALGEBRA OF LIMITS AT INFINITY

Theorem 4.4.23 Suppose lim f(x) = L, lim g(x) = M, and c E IR. Then

x->+oo x->+oo

(a ) lim (cf(x )) = cL.

x ->+oo

(b) lim (f(x ) ± g(x)) = L ± M;

x->+oo

(c) lim (f(x )g(x )) = LM;

x->+oo

(d) lim (f(x)) = ..£ (provided M =f=. 0).

x->+oo g(x) M

(e) Squeeze Principle: if lim f(x) = lim g(x) = L , and \fx in some

x->+oo x ->+oo

neighborhood of +oo, f(x) S h(x) s g(x), then lim h(x) = L.

x->+oo

(f) Limits Preserve Inequalities: if lim f(x) and lim g(x) exist and

x->+oo x ->+oo

f(x) S g( x ) in some n eighborhood of + oo, then lim f(x) S lim g(x).

x->+oo x ->+oo

The above results remain true if +oo is replaced by -oo. They also remain
true if L and M are replaced by +oo or -oo, in the sense described by Tables
4.1, 4.2, and 4.3 of this section.

Proof. Exercise 17 (Project). •