Calculus: Analytic Geometry and Calculus, with Vectors

482 Exponential and logarithmic functions

Theorem 9.15 .ds a function of the "rational variable" x, the function

having values ax is continuous over - o < x < -.

We shall prove this theorem with the aid of (9.13) and an estimate of

the troublesome factor (ah - 1) that is, from our present point of view

and from some others, quite amazing.

o h 2h 3 i x Let h be a positive rational number for

which h 1 and let n be the greatest

Figure 9.151 integer for which nh 5 1. If we wish,

we can draw Figure 9.151 and put a

part of the graph of y = ax over it to help us see what we are doing. The

equality

(9.152) (ah - 1)a(k-1)h = akh - a(k-1)h

holds when k 1, 2, 3, , nand summing over these values of k gives

the formula

rT+ ny

(9.153) (ah - 1) L ) (akh - a(k-1)h)

G

a(k-1)h a

k-1 k-1

= (ah - a°) + (ash - ah) + (ash - a2h)+. + (anh - a(n-1)h)

=anh-a°=anh-1 <a-1

which has, in its middle line, a sum which is called a telescopic sum because

it "telescopes" to anh - a°. More critical examination of the sum in

brackets will appear later. Meanwhile, we observe that if we denote

the sum by S, then each term in the sum is 1 or more, so S >= n. But

nh > -J, so n > 1/2h and S > 1/2h. Replacing S in (9.153) by the

smaller number 1/2h gives the inequality

(ah-1)2h<a-1.

Therefore,

(9.154) ah-1<2(a-1)h (0<h<1).

This and (9.13) show that the formula

(9.155) av Fh - ax < 2(a - 1)hax

is valid when x and h are rational and 0 < h < 1. This result and the

fact that ax is increasing give the following theorem.

Theorem 9.16 If r1 and r2 are rational numbers for which

Ire-r1l =h51

and if M is a rational number for which r1 < M and r2 < M, then

la" - aril 5 2(a - 1)jr2 - r1jaM.

This is a stronger version of Theorem 9.15 which implies Theorem 9.15.

Our information about the values of ar for rational numbers r is now