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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
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124 Chapter 1 • Limits and Continuity
Teaching Tip
Section objective 4: Determine where
an exponential or a logarithmic function
is continuous is a good review. This
knowledge is a calculus prerequisite, so if
the students are well prepared, you may
be able to address this objective briefly to
save time.
4 Determine Where an Exponential or a Logarithmic
Function Is Continuous
The graphs of an exponential function y = a x and its inverse function y = log a x are
shown in Figure 46. The graphs suggest that an exponential function and a logarithmic
function are continuous on their domains. We state the following theorem without
proof.
y a x
1
(1, )
a
y
3
(0, 1)
(a, 1)
(1, a)
y x
y y a x
3
(1, a)
(0, 1) 1
(1, )
a
(a, 1)
(1, 0)
y x
y log a x
Students should be familiar with the
graphs of y = e x and y = ln x, so that
they can easily evaluate the following
limits:
x
lim e = 0
x→−∞
x
lim e = 1
x→0
x
lim e
x→∞
=∞
lim lnx
=−∞
+
x→0
lim lnx
= 0
x→1
lim lnx
=∞
x→∞
AP® Exam Tip
AP® CaLC skill builder
for example 5
Show a Composite Function Is
Continuous
Which of the following functions is (are)
continuous at x = 0?
2
x /2
I. fx ( ) = e
II. gx ( ) =
3 cos x
III. hx ( ) = e −2x
tanx
Solution
2
x /2
I. fx ( ) = e is continuous for all real
numbers, so the function is
continuous at x = 0.
II. gx ( ) = 3 cos x is continuous on its
domain, the set of all real numbers.
Since the composition of two
continuous functions is a continuous
function, g is continuous at x = 0.
III. The function hx ( ) = e −2x
tanx
is
defined and continuous on
π π
− < x < , so h is continuous at
2 2
x = 0.
CALC
CLIP
3 (1, 0)
3 x
Figure 46
3
(a) 0 a 1
1
( , 1)
a
y log a x
3 3
3
(b) a 1
1
( , 1)
a
THEOREM Continuity of Exponential and Logarithmic Functions
• An exponential function is continuous on its domain, all real numbers.
• A logarithmic function is continuous on its domain, all positive real numbers.
Based on this theorem, the following two limits can be added to the list of basic
limits.
lim
x→c ax = a c for all real numbers c, a > 0, a = 1
and
lim log
x→c
a x = log a c for any real number c > 0, a > 0, a = 1
EXAMPLE 5
Show that:
Showing a Composite Function Is Continuous
(a) f (x) = e 2x is continuous for all real numbers.
(b) F(x) = 3√ ln x is continuous for x > 0.
Solution (a) The domain of the exponential function is the set of all real numbers, so
f is defined for any number c. That is, f (c) = e 2c . Also for any number c,
[
lim f (x) = lim
x→c x→c e2x = lim(e x ) 2 = lim ex] 2
= (e c ) 2 = e 2c = f (c)
x→c x→c
Since lim f (x) = f (c) for any number c, then f is continuous at all numbers c.
x→c
(b) The logarithmic function f (x) = ln x is continuous on its domain, the set of
all positive real numbers. The function g(x) = 3√ x is continuous on its domain, the
set of all real numbers. Then for any real number c > 0, the composite function
F(x) = (g ◦ f )(x) = 3√ ln x is continuous at c. That is, F is continuous at all real
numbers x > 0. ■
x
124
Chapter 1 • Limits and Continuity
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