Chapter 3
Limits and the
Derivative
Section 3
Continuity
(Part 1)
Learning Objectives for Section 3.3
Continuity
 The student will
understand the concept
of continuity
 The student will be able
to apply the continuity
properties
 The student will be able
to solve word problems.
 The student will be able
to solve inequalities
Barnett/Ziegler/Byleen Business Calculus 12e
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Continuity
In this lesson, we’ll take a closer look at graphs that are
discontinuous due to:
• Holes
• Gaps
• Asymptotes
Barnett/Ziegler/Byleen Business Calculus 12e
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Definition of Continuity
A function f is continuous at a point x = c if it meets
these three criteria:
1. lim 𝑓(𝑥) ≠ 𝐷𝑁𝐸
𝑥→𝑐
2. f (c)  undefined
3. lim f ( x)  f (c)
x c
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 1
Is f(x) continuous at x = 2 ?
?
lim 𝑓 𝑥 = 𝑓(2)
𝑥→2
𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝐷𝑁𝐸
f(x) is not continuous at x = 2
Is f(x) continuous at x = 3 ?
?
lim 𝑓 𝑥 = 𝑓(3)
𝑥→3
1 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
f(x) is not continuous at x = 3
Continuous over the interval:
Barnett/Ziegler/Byleen Business Calculus 12e
−∞, 2 ∪ (2, 3) ∪ (3, ∞)
5
Example 2
Is f(x) continuous at x = -2 ?
?
lim 𝑓 𝑥 = 𝑓(−2)
𝑥→−2
3
−1
f(x) is not continuous at x = -2
Continuous over the interval:
Barnett/Ziegler/Byleen Business Calculus 12e
−∞, −2 ∪ (−2, ∞)
6
Example 3
𝑓 𝑥 = 2𝑥 2 + 𝑥 − 1 Is f(x) continuous at x = 3?
?
lim 𝑓 𝑥 = 𝑓(3)
𝑥→3
20
20
f(x) is continuous at x = 3
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 4
𝑥2 − 4
𝑓 𝑥 =
𝑥+2
Is f(x) continuous at x = -2?
𝑥2 − 4 ?
lim
= 𝑓(−2)
𝑥→−2 𝑥 + 2
(𝑥 + 2)(𝑥 − 2)
𝑥2 − 4
lim
1. lim
= 𝑥→−2
𝑥+2
𝑥→−2 𝑥 + 2
= lim (𝑥 − 2) = −4
𝑥→−2
2. 𝑓 −2 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
f(x) is NOT continuous at x = -2
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 5
𝑥−5
𝑓 𝑥 =
𝑥−5
Is f(x) continuous at x = 5? Explain.
𝑥−5 ?
lim
= 𝑓(5)
𝑥→5 𝑥 − 5
−(𝑥 − 5)
𝑥−5
= −1
lim−
= lim−
𝑥→5
𝑥→5 𝑥 − 5
𝑥−5
𝑥−5
1. lim
(𝑥 − 5)
𝑥−5
𝑥→5 𝑥 − 5
= 1
lim+
= lim+
𝑥→5
𝑥−5
𝑥→5 𝑥 − 5
lim 𝑓 𝑥 = 𝐷𝑁𝐸
𝑥→5
2. 𝑓 5 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
f(x) is NOT continuous at x = 5
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 6
𝑥−5
𝑓 𝑥 =
𝑥−5
Is f(x) continuous at x = 2?
𝑥−5 ?
lim
= 𝑓(2)
𝑥→2 𝑥 − 5
𝑥−5
1. lim
=
𝑥→2 𝑥 − 5
2−5
= −1
2−5
2. 𝑓 2 = −1
f(x) is continuous at x = 2
Barnett/Ziegler/Byleen Business Calculus 12e
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Continuity
 Where is a graph continuous?
• Where there are no asymptotes or holes.
• Where the function is defined.
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 7
 Where is 𝑓 𝑥 = 2𝑥 + 6 continuous?
2𝑥 + 6 ≥ 0
𝑥 ≥ −3
 𝑓(𝑥) is continuous over the interval: [−3 , ∞)
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 8
 Where is 𝑓 𝑥 =
(𝑥+1)(𝑥−4)
(𝑥+1)(𝑥−2)
continuous?
x ≠ −1, 2
 𝑓(𝑥) is continuous over the interval:
(−∞, −1) ∪ (−1,2) ∪ (2 , ∞)
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 9
 Where is 𝑓 𝑥 continuous?
𝑓(𝑥) is continuous over the interval:
(−∞, −2) ∪ (−2,1) ∪ (1,2) ∪ (2 , ∞)
Barnett/Ziegler/Byleen Business Calculus 12e
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Homework
#3-3A
Pg 161
(7-14, 23, 27,
29, 31, 49-59 odd)
Barnett/Ziegler/Byleen Business Calculus 12e
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Chapter 3
Limits and the
Derivative
Section 3
Continuity
(Part 2)
Learning Objectives for Section 3.3
Continuity
 The student will
understand the concept
of continuity
 The student will be able
to apply the continuity
properties
 The student will be able
to solve word problems.
 The student will be able
to solve inequalities
Barnett/Ziegler/Byleen Business Calculus 12e
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Application: Media
 A music website called MyTunes charges $0.99 per song if
you download less than 100 songs per month and $0.89 per
song if you download 100 or more songs per month.
 Write a piecewise function f(x) for the cost of
downloading x songs per month.
 Graph the function.
 Is f(x) continuous at x = 100? Explain.
Barnett/Ziegler/Byleen Business Calculus 12e
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Solution
𝑓 𝑥 =
0.99𝑥 0 ≤ 𝑥 < 100
0.89𝑥
𝑥 ≥ 100
$99
?
lim 𝑓 𝑥 = 𝑓(100)
𝑥→100
𝐷𝑁𝐸
$89
89
$79
100
110
Barnett/Ziegler/Byleen Business Calculus 12e
f(x) is NOT continuous at x=100
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Application: Natural Gas Rates
 The table shows the monthly rates for natural gas charged
by MyGas Company. The charge is based on the number of
therms used per month. (1 therm = 100,000 Btu)
Amount
Cost
Base Charge
$8.00
First 40 therms
$0.60 per therm
Over 40 therms
$0.35 per therm
 Write a piecewise function f(x) of the monthly charge for x
therms.
 Graph f(x).
 Is f(x) continuous at x = 40? Give a mathematical reason.
Barnett/Ziegler/Byleen Business Calculus 12e
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Solution
Amount
Cost
Base Charge
$8.00
First 40 therms
$0.60 per therm
Over 40 therms
$0.35 per therm
8 + 0.60𝑥
0 ≤ 𝑥 ≤ 40
𝑓 𝑥 =
8 + .60 40 + .35(𝑥 − 40)
𝑥 > 40
=
8 + 0.60𝑥
.35𝑥 + 18
Barnett/Ziegler/Byleen Business Calculus 12e
0 ≤ 𝑥 ≤ 40
𝑥 > 40
21
Solution
𝑓(𝑥) =
8 + 0.60𝑥
.35𝑥 + 18
0 ≤ 𝑥 ≤ 40
𝑥 > 40
?
lim 𝑓 𝑥 = 𝑓(40)
$90
𝑥→40
$60
32
$30
40
80
Barnett/Ziegler/Byleen Business Calculus 12e
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f(x) IS continuous at x=40
22
Solving Inequalities
 Up until now, we have solved inequalities using a
graphical approach.
• 𝑓 𝑥 < 0  Where is the graph below the x-axis?
• 𝑓 𝑥 > 0  Where is the graph above the x-axis?
 Now we will learn an algebraic approach that is based on
continuity properties.
Barnett/Ziegler/Byleen Business Calculus 12e
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Constructing Sign Charts
1. Find all numbers which are:
a. Holes or vertical asymptotes.
Plot these as open circles on the number line.
b. x-intercepts
Plot these according to the inequality symbol.
2. Select a test number in each interval and determine if f (x)
is positive (+) or negative (–) in the interval.
3. Determine your answer using the signs and the inequality
symbol and write it using interval notation.
Barnett/Ziegler/Byleen Business Calculus 12e
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Polynomial Inequalities
 Ex 3: Solve and write your answer in interval notation.
𝑥 2 − 4𝑥 − 12 > 0
Factor f(x) and solve for zeros.
(x − 6)(x + 2) > 0
Graph the zeros on a number line.
Use open circles for < or >, use
closed circles for  or .
+
-
+
−2
6
𝐴𝑛𝑠𝑤𝑒𝑟: (−∞, −2) ∪ (6, ∞)
Test numbers on all sides of the
zeros by plugging them into the
inequality.
Since f(x) > 0 we want the positive
intervals.
Barnett/Ziegler/Byleen Business Calculus 12e
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Rational Inequalities
 Ex 4: Solve and write your answer in interval notation.
𝑥2 − 4
Factor the top and bottom.
≤0
𝑥+4
Graph the holes and vertical asymptotes as
(𝑥 + 2)(𝑥 − 2)
≤0
𝑥+4
−
−4
+
+
−
−2
2
open circles.
Graph the x-intercepts. Use open circles for <
or >, use closed circles for  or .
Test numbers on all sides of the
points by plugging them into the
reduced inequality.
𝐴𝑛𝑠𝑤𝑒𝑟: −∞, −4 ∪ [−2,2] Since f(x)  0, we want the negative
intervals.
Barnett/Ziegler/Byleen Business Calculus 12e
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Homework
Barnett/Ziegler/Byleen Business Calculus 12e
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