1549901369-Elements_of_Real_Analysis__Denlinger_

5.6 *Exponentials, Powers, and Logarithms 281

(3) Let x E JR. In Definition 5.6.5, the sequence { arn} is a monotone in-

creasing sequence of positive numbers, so ax = lim arn > 0. Thus, f(x) =ax

is positive-valued.

(4) We know that lim an

n--+oo

f(x) =ax has no upper bound.

n-HXl

+oo (Example 2.4.6). Thus, the range of

Corollary 5.6.7 (a) If a> 1, then lim ax = +oo and lim ax= 0.

x-++oo x-+-oo

(b) If 0 <a< 1, then lim ax= 0 and lim ax= +oo.

x-++oo x-+-oo

Proof. Exercise 2. •

We would also like to prove that the exponential function f(x) = ax is
continuous everywhere on JR. The proof of that, however, must wait until we
establish a few more properties of this function.

Theorem 5.6.8 (Algebraic Properties of Exponents) Let a, b > 0. The
exponential function f(x) =ax satisfies the following algebraic properties:

(a) a^0 = 1

(b) axay = ax+y

(c) ax /aY = ax-y

(d) (ab)X = axbx

(e) a-x = (ax)-1 = (a-l)x

(f) (a/b)X =ax /bx

Proof. (a) Use the constant sequence {rn} = {O} in Definition 5.6.5.
(b) Let x, y E JR. Then :3 monotone increasing sequences { r n}, {Sn} of
rational numbers :::i rn --> x and Sn --> y. Then {rn + sn} is a monotone
increasing sequence of rational numbers converging to x + y, so

axay = ( lim arn) ( lim a•n) = lim (arna•n)

n-+oo n--+oo n-+oo

= lim (arn+sn) (by Exercise 5.5.15)

n--+oo

(c) Exercise 3.
(d) Let a, b E JR. Let {rn} be a monotone increasing sequence of rational
numbers converging to x. Then

(ab)X = lim (abrn = lim (arnbrn) (by Exercise 5.5.15)

= (]; arn) u~ri~rn)

= axbx.

( e) Exercise 4.

( f) Exercise 5. •