286 Chapter 5 11 Continuous Functions
Theorem 5.6.15 (Negative Power Functions) Let t < 0. The power
function f ( x) = xt, defined with the help of Definition 5. 6. 5, is strictly de-
creasing, positive, and continuous everywhere on (0, + oo), lim xt = 0, and
X->00
lim xt = +oo.
x->O+
Proof. Exercise 9. •Theorem 5.6.16 (A Further Algebraic Property of Exponents) Let a>
- Then 'Vx, y E rr:t, ( a x )y = axy.
Proof. Part l. We first prove t he result when xis rational; call it x = r E Q.
Let y E R Let {Sn} be a sequence of rational numbers 3 Sn --+ y. Then, by
Definition 5.6.5,
( ar)Y = lim ( ar)8n.
n->ooAlso rsn --+ ry so by Definition 5.6.5 and Exercise 5.5.15,
ary = lim aTSn = lim (ar)8n.
n~oo n~ooTherefore, ary = ( ar)Y.
P art 2. Let y E rr:t and suppose x is any real number. Then 3 sequen ce
{rn} of rational numbers 3 rn--+ x. Then, by Definition 5.6.5, arn --+ax, so by
Theorem 5.6.14,
On the other hand, rny--+ xy, so by Theorem 5.6.10,e AND e x AS LIMITSIn Section 2.5 we proved that the sequential limit nlim ~oo (1 + .!.)n n exists; we
called this number e. We now take a look at several function limits that also
equal e.
Theorem 5.6. 17 lim (1 + l)x = e.
x-.oo x
The first thing to see is that there is something to prove. Notice that the
limit described here is not the limit of a sequence, but the limit of a function.
Our task will be to make the transition from the sequence { (1 + ~r}to the