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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
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104 Chapter 1 • Limits and Continuity
AP® CaLC skill builder
for example 2
Determining Whether a Function Is
Continuous at a Number
3
x −25x
Determine whether f( x)
= is
2
continuous at x = 5. x −5x
Solution
We first check the conditions for continuity.
To determine if 5 is in the domain of f,
compute
3
5 −25(5)
f (5) =
2
5 −5(5)
This shows that f (5) is not defined,
because both numerator and denominator
are 0. Since 5 is not in the domain of f,
then f is not continuous at 5.
AP® CaLC skill builder
for example 2
Determining Whether a Function Is
Continuous at a Number
4
( x −1)
Use the graph of f( x)
= to
1−
x
determine whether f is continuous at
x = 1. If f is discontinuous, what type of
discontinuity does the function have?
y
y
4
2
x
f(x) 2 2
x 2 4
4 2 2 4
2
(0,
4 )
2
4
Figure 23 f is continuous at 0;
f is discontinuous at −2 and 2.
x
Figure 22 shows the graphs of f and g from Example 1. Notice that f is continuous
at 1, and its graph is drawn without lifting the pencil. But the function g is discontinuous
at 2, and to draw its graph, you must lift your pencil at x = 2.
10
f (x) 3x 2 5x 4
4
Figure 22
2
y
20
15
5
(1, 2)
2
4
x
(a) f is continuous at 1. (b) g is discontinuous at 2.
4
2
y
10
5
5
2
g(x) x2 9
x 2 4
NOW WORK Problem 19 and AP® Practice Problem 3.
Determining Whether a Function Is Continuous
EXAMPLE 2 at a Number
x
2
+ 2
Determine if f (x) = is continuous at the numbers −2, 0, and 2.
x 2 − 4
Solution The domain of f is {x|x = −2, x = 2}. Since f is not defined at −2 and 2,
the function f is not continuous at −2 and at 2. The number 0 is in the domain of f.
√
2
That is, f is defined at 0, and f (0) =−
4 . Also,
x
2
+ 2
lim f (x) = lim
x→0 x→0 x 2 − 4
lim x 2
+ 2 lim
= x→0
lim (x 2 − 4) = lim
x→0
x→0
(x 2 + 2)
x→0 x 2 − lim
x→0
4
√ √
0 + 2 2
=
0 − 4 =− 4 = f (0)
The three conditions of continuity at a number are met. So, the function f is continuous
at 0. ■
Figure 23 shows the graph of f.
NOW WORK Problem 21.
4
x
4
24
2
22
22
24
2
4
x
CALC
CLIP
EXAMPLE 3
Determining Whether a Piecewise-Defined Function
Is Continuous
Determine whether the function
⎧
x 2 − 9
if x < 3
⎪⎨ x − 3
f (x) =
9 if x = 3
⎪⎩
x 2 − 3 if x > 3
is continuous at 3.
Solution
The graph of f reveals a hole at
x = 1 because f (1) is undefined. We can
conclude that f is discontinuous at x = 1.
Since lim − fx ( ) = lim + fx ( ) =−4,
x→1 x→1
the function has a removable discontinuity
at x = 1. If we define f (1) = − 4, then f is a
continuous function.
TRM Section 1.3: Worksheet 1
This worksheet contains 7 functions along
with their graphs. Students are asked to
identify the point(s) of discontinuity as
well as to state why the listed points are
discontinuous.
Calculator Tip
Some students may be tempted to graph functions
on their calculator to make decisions about
continuity. Most students, though, use a default
calculator setting in which holes cannot be seen
on the graph that is made on the calculator screen.
This might lead the students to (falsely) believe
a function is continuous when it is not. Students
mainly should determine points of discontinuity
analytically and then use a graph to support their
conclusion.
104
Chapter 1 • Limits and Continuity
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