Test Bank for Precalculus, 6th Edition

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Test Bank for Precalculus, 6th Edition helps you familiarize yourself with exam formats and key concepts.

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Ch. 11 Introduction to Calculus
11.1 Finding Limits Using Tables and Graphs
1 Understand Limit Notation
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Choose the table which contains the best values of x for finding the requested limit of the given function.
1) lim
x→2
(x2 + 8x - 2)
A) x 1.9 1.99 1.999 2.001 2.01 2.1
f(x)
B) x 0.9 0.99 0.999 1.001 1.01 1.1
f(x)
C) x -1.9 -1.99 -1.999 2.001 2.01 2.1
f(x)
D) x -0.9 -0.99 -0.999 1.001 1.01 1.1
f(x)
2) lim
x→1
x4 - 1
x - 1
A) x 0.9 0.99 0.999 1.001 1.01 1.1
f(x)
B) x 1.9 1.99 1.999 1.001 1.01 1.1
f(x)
C) x -0.9 -0.99 -0.999 -1.001 -1.01 -1.1
f(x)
D) x 0.1 0.19 0.119 1.99 1.09 1.9
f(x)
Page 1

3) lim
x→0
sin 2x
x
A) x -0.03 -0.02 -0.01 0.01 0.02 0.03
f(x)
B) x -0.01 -0.02 -0.03 0.03 0.02 0.01
f(x)
C) x -0.3 -0.2 -0.1 0.1 0.2 0.3
f(x)
D) x 0.03 0.02 0.01 0.001 0.002 0.003
f(x)
Translate the given limit notation into a sentence.
4) lim
x→4
( x - 2) = 0
A) The limit of x - 2 as x approaches 4 equals the number 0.
B) The limit of x - 2 as x approaches 0 equals the number 4.
C) The limit of x - 4 as x approaches 2 equals the number 0.
D) The limit of x - 4 as x approaches 0 equals the number 2.
5) lim
x→0
sin 2x
x = 2
A) The limit of sin 2x
x as x approaches 0 equals the number 2.
B) The limit of sin 2x
x as x approaches 2 equals the number 0.
C) The limit of sin 2x
x as x approaches 0 from the left equals the number 2.
D) The limit of sin 2x
x as x approaches 0 from the right equals the number 2.
2 Find Limits Using Tables
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Complete the table for the function and find the indicated limit.
1) lim
x→2
(x2 + 8x - 2)
x 1.9 1.99 1.999 2.001 2.01 2.1
f(x)
A) 16.810; 17.880; 17.988; 18.012; 18.120; 19.210
limit = 18.0
B) 5.043; 5.364; 5.396; 5.404; 5.436; 5.763
limit = 5.40
C) 16.692; 17.592; 17.689; 17.710; 17.808; 18.789
limit = 17.70
D) 6.810; 7.880; 7.988; 8.012; 8.120; 9.210
limit = 8.0
Page 2

2) lim
x→1
x4 - 1
x - 1
x 0.9 0.99 0.999 1.001 1.01 1.1
f(x)
A) 3.439; 3.940; 3.994; 4.006; 4.060; 4.641
limit = 4.0
B) 1.032; 1.182; 1.198; 1.201; 1.218; 1.392
limit = 1.210
C) 4.595; 5.046; 5.095; 5.105; 5.154; 5.677
limit = 5.10
D) 7.439; 7.940; 7.994; 8.006; 8.060; 8.641
limit = 8.0
3) lim
x→0
x3 - 6x + 8
x - 2
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x)
A) -4.09476; -4.00995; -4.00100; -3.99900; -3.98995; -3.89526
limit = -4.0
B) -1.22843; -1.20298; -1.20030; -1.19970; -1.19699; -1.16858
limit = -1.20
C) -2.18529; -2.10895; -2.10090; -2.09910; -2.09096; -2.00574
limit = -2.10
D) 4.09476; 4.00995; 4.00100; 3.99900; 3.98995; 3.89526
limit = 4.0
4) lim
x→0
(x2 - 5)
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x)
A) -4.9900; -4.9999; -5.0000; -5.0000; -4.9999; -4.9900
limit = -5.0
B) -1.4970; -1.4999; -1.5000; -1.5000; -1.4999; -1.4970
limit = -1.50
C) -2.9910; -2.9999; -3.0000; -3.0000; -2.9999; -2.9910
limit = -3.0
D) 1.4970; 1.4999; 1.5000; 1.5000; 1.4999; 1.4970
limit = 1.50
5) lim
x→0
sin 2x
x
x -0.03 -0.02 -0.01 0.01 0.02 0.03
f(x)
A) 1.9988, 1.9995, 1.9999, 1.9999, 1.9995, 1.9988
limit = 2
B) 1.9988, 1.9995, 1.9999, 1.9999, 1.9995, 1.9988
limit = 0
C) 1.9988, 1.9995, 1.9999, 1.9999, 1.9995, 1.9988
limit = 1
D) 1.9988, 1.9995, 1.9999, 1.9999, 1.9995, 1.9988
limit = -2
Page 3

6) lim
x→0
x2
sin x
x -0.03 -0.02 -0.01 0.01 0.02 0.03
f(x)
A) -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300
limit = 0
B) -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300
limit = 1
C) -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300
limit = 0.1
D) -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300
limit = -1
7) lim
x→ -7
x2 - 49
x + 7
x -7.1 -7.01 -7.001 -6.999 -6.99 -6.9
f(x)
A) -14.1, -14.01, -14.001, -13.999, -13.99, -13.9
limit = -14
B) 7.1, 7.01, 7.001, 6.999, 6.99, 6.9
limit = 7
C) -14.1, -14.01, -14.001, -13.999, -13.99, -13.9
limit = -14
D) -7.1, -7.01, -7.001, -6.999, -6.99, -6.9
limit = -7
8) lim
x→0
1 - cos x
sin x
x -0.03 -0.02 -0.01 0.01 0.02 0.03
f(x)
A) -0.0150, -0.0100, -0.0050, 0.0050, 0.0100, 0.0150
limit = 0
B) -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300
limit = 1
C) -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300
limit = 0.1
D) -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300
limit = -1
9) lim
x→0
f(x), where f(x) = x + 5 if x < 0
3x + 5 if x ≥ 0
x -0.1 -0.01 -0.001 0.001 0.01 0.1
f(x)
A) 4.9, 4.99, 4.999, 5.003, 5.03, 5.3
limit = 5
B) 2.9, 2.99, 2.999, 3.003, 3.03, 3.3
limit = 3
C) 0.9, 0.99, 0.999, 1.003, 1.03, 1.3
limit = 1
D) 1.9, 1.99, 1.999, 2.003, 2.03, 2.3
limit = 2
Page 4

3 Find Limits Using Graphs
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit
or function value does not exist.
1) a. lim
x→0
f(x) b. f(0)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
4
3
2
1
-1
-2
-3
-4
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y
4
3
2
1
-1
-2
-3
-4
A) a. lim
x→0
f(x) = 0
b. f(0) = 0
B) a. lim
x→0
f(x) = 0
b. f(0) = 2
C) a. lim
x→0
f(x) = -2
b. f(0) = -2
D) a. lim
x→0
f(x) does not exist
b. f(0) does not exist
2) a. lim
x→0
f(x) b. f(0)
x-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
y
1
-1
x-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
y
1
-1
A) a. lim
x→0
f(x) = 1
b. f(0) = 1
B) a. lim
x→0
f(x) = 0
b. f(0) = 0
C) a. lim
x→0
f(x) = -1
b. f(0) = -1
D) a. lim
x→0
f(x) does not exist
b. f(0) does not exist
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3) a. lim
x→0
f(x) b. f(0)
A) a. lim
x→0
f(x) = 1
b. f(0) does not exist
B) a. lim
x→0
f(x) = -1
b. f(0) does not exist
C) a. lim
x→0
f(x) = 0
b. f(0) = 0
D) a. lim
x→0
f(x) does not exist
b. f(0) does not exist
4) a. lim
x→0
f(x) b. f(0)
x-2 -1 1 2 3 4 5
y
10
8
6
4
2
-2
x-2 -1 1 2 3 4 5
y
10
8
6
4
2
-2
A) a. lim
x→0
f(x) = 0
b. f(0) = 6
B) a. lim
x→0
f(x) = 6
b. f(0) = 6
C) a. lim
x→0
f(x) = 6
b. f(0) = 0
D) a. lim
x→0
f(x) does not exist
b. f(0) = 6
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5) a. lim
x→0
f(x) b. f(0)
A) a. lim
x→0
f(x) = 0
b. f(0) does not exist
B) a. lim
x→0
f(x) = 1
b. f(0) = 1
C) a. lim
x→0
f(x) = -1
b. f(0) does not exist
D) a. lim
x→0
f(x) does not exist
b. f(0) does not exist
6) a. lim
x → 1
f(x) b. f(1)
x-3 -2 -1 1 2
y2
1
-1
-2
-3
-4
x-3 -2 -1 1 2
y2
1
-1
-2
-3
-4
A) a. lim
x → 1
f(x) = -3
b. f(1) = -3
B) a. lim
x → 1
f(x) = 3
b. f(1) = 3
C) a. lim
x → 1
f(x) = 1
b. f(1) = 1
D) a. lim
x → 1
f(x) does not exist
b. f(1) = -3
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7) a. lim
x→1
f(x) b. f(1)
x1 2 3 4 5
y
4
3
2
1
-1
x1 2 3 4 5
y
4
3
2
1
-1
A) a. lim
x→1
f(x) = 3
b. f(1) = 3
B) a. lim
x→1
f(x) = 1
b. f(1) = 3
C) a. lim
x→1
f(x) = 0
b. f(1) = 0
D) a. lim
x→1
f(x) does not exist
b. f(1) does not exist
8) a. lim
x→2
f(x) b. f(2)
x1 2 3 4 5
y
5
4
3
2
1
x1 2 3 4 5
y
5
4
3
2
1
A) a. lim
x→2
f(x) = 2
b. f(2) = 1
B) a. lim
x→2
f(x) = 1
b. f(2) = 1
C) a. lim
x→2
f(x) = 1
b. f(2) = 2
D) a. lim
x→2
f(x) does not exist
b. f(2) = 1
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9) a. lim
x → 1
f(x) b. f(1)
x-3 -2 -1 1 2
y2
1
-1
-2
-3
-4
x-3 -2 -1 1 2
y2
1
-1
-2
-3
-4
A) a. lim
x → 1
f(x) = -3
b. f(1) = 1
B) a. lim
x → 1
f(x) = -3
b. (1) = -3
C) a. lim
x → 1
f(x) = 1
b. f(1) = -3
D) a. lim
x → 1
f(x) = -3
b. f(1) does not exist
Use the graph and the viewing rectangle shown below the graph to find the indicated limit.
10) lim
x→-2
(x2 - 2)
[-6, 6, 1] by [-6, 6, 1]
A) lim
x→-2
(x2 - 2) = 2 B) lim
x→-2
(x2 - 2) = -6
C) lim
x→-2
(x2 - 2) = 6 D) lim
x→-2
(x2 - 2) = -2
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4 Find One-Sided Limits and Use Them to Determine If a Limit Exists
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit
or function value does not exist.
1) a. lim
x→3
f(x) b. f(3)
x1 2 3 4 5
y
5
4
3
2
1
x1 2 3 4 5
y
5
4
3
2
1
A) a. lim
x→3
f(x) does not exist
b. f(3) = 5
B) a. lim
x→3
f(x) = 5
b. f(3) = 5
C) a. lim
x→3
f(x) = 3
b. f(3) = 5
D) a. lim
x→3
f(x) = 4
b. f(3) does not exist
2) a. lim
x→1
f(x) b. f(1)
x-2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
x-2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
A) a. lim
x→1
f(x) does not exist
b. f(1) = 2
B) a. lim
x→1
f(x) = 1
b. f(1) = 2
C) a. lim
x→1
f(x) = 2
b. f(1) = 2
D) a. lim
x→1
f(x) = 1
b. f(1) = 0
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5) a. lim
x→-1/2
f(x) b. f(-1/2)
A) a. lim
x→-1/2
f(x) = -1
b. f(-1/2) = -1
B) a. lim
x→-1/2
f(x) = -2
b. f(-1/2) = -1
C) a. lim
x→-1/2
f(x) = 0
b. f(-1/2) does not exist
D) a. lim
x→-1/2
f(x) does not exist
b. f(-1/2) = -1
6) a. lim
x→3-
f(x) b. f(3)
x1 2 3 4 5
y
5
4
3
2
1
x1 2 3 4 5
y
5
4
3
2
1
A) a. lim
x→3-
f(x) = 3
b. f(3) = 5
B) a. lim
x→3-
f(x) = 3
b. f(3) = 3
C) a. lim
x→3-
f(x) = 3
b. f(3) does not exist
D) a. lim
x→3-
f(x) does not exist
b. f(3) = 5
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9) a. lim
x→-1-
f(x) b. f(-1)
A) a. lim
x→-1-
f(x) = -1
b. f(-1) does not exist
B) a. lim
x→-1-
f(x) = 0
b. f(-1) = 1
C) a. lim
x→-1-
f(x) = -1
b. f(-1) = 1
D) a. lim
x→-1-
f(x) does not exist
b. f(-1) does not exist
10) a. lim
x→-2+
f(x) b. f(-2)
A) a. lim
x→-2+
f(x) = -2
b. f(-2) = -2
B) a. lim
x→-2+
f(x) = -1
b. f(-2) does not exist
C) a. lim
x→-2+
f(x) = -2
b. f(-2) = -1
D) a. lim
x→-2+
f(x) does not exist
b. f(-2) = -1
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11) a. lim
x→2.5-
f(x) b. f(2.5)
A) a. lim
x→2.5-
f(x) = 1
b. f(2.5) = 1
B) a. lim
x→2.5-
f(x) = 2
b. f(2.5) = 1
C) a. lim
x→2.5-
f(x) = 0
b. f(2.5) = 2
D) a. lim
x→2.5-
f(x) does not exist
b. f(2.5) = 2
12) a. lim
x→1-
f(x) b. f(1)
x-2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
x-2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
A) a. lim
x→1-
f(x) = 1
b. f(1) = 2
B) a. lim
x→1-
f(x) = 2
b. f(1) = 2
C) a. lim
x→1-
f(x) = 1
b. f(1) = 1
D) a. lim
x→1-
f(x) does not exist
b. f(1) = 2

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5 Solve Apps: Use Limits
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) You are riding your motorcycle along a freeway traveling x miles per hour. The function
f(x) = 0.0014x2 + x + 9 describes the recommended safe distance, f(x), in feet, between your motorcycle and
the other motorcycles on the freeway. Use the values in the table below to solve this exercise.
x 49.9 49.99 49.999 → ← 50.001 50.01 50.1
f(x) 93.76 93.976 93.998 → ← 94.002 94.024 94.24
Find lim
x→50
f(x). Describe what this means in terms of your motorcycle's speed and the recommended safe
distance.
A) 94; This means that the recommended safe distance between motorcycles traveling at 50 miles per
hour is 94 feet.
B) 50; This means that the recommended safe distance between motorcycles traveling at 50 miles per
hour is 50 feet.
C) 93.999; This means that the recommended safe distance between motorcycles traveling at 50 miles per
hour is 93.999 feet.
D) lim
x→ 50
f(x) does not exist.
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2) You rent a moving van from a company that charges $10 per hour plus $0.50 per mile. The van is driven 40
miles in the first hour. The figure below shows the graph of the cost, f(x), in dollars, as a function of the
miles, x, that you drive the van.
(i) Find lim
x→ 20
f(x). Interpret the limit, referring to miles driven and cost.
(ii) For the first hour hour only, what is the rental cost approaching as the mileage gets closer to 40?
(iii) What is the cost to rent the van at the start of the second hour?
A) (i) 20; the cost to rent the van for one hour and drive it 20 miles is $20.
(ii) $30
(iii) $40
B) (i) 30; the cost to rent the van for one hour and drive it 20 miles is $30.
(ii) $40
(iii) $40
C) (i) 40; the cost to rent the van for one hour and drive it 20 miles is $40.
(ii) $30
(iii) $40
D) (i) 20; the cost to rent the van for one hour and drive it 20 miles is $20.
(ii) $30
(iii) $30
3) You are building a screened-in porch attached to your house. Because the house will be used for one side
of the enclosure, only three sides will need to be enclosed. You have 24 yards of material to enclose the
three walls. The function f(x) = x(24 - 2x) describes the area of the screened-in porch that you can enclose,
f(x), in square yards, if the width of the screened-in porch is x yards
X Y1
5.7
5.8
5.9
6
6.1
6.2
6.3
71.82
71.92
71.98
71.98
71.92
71.82
Y1 = X * (24 - 2X)
(i) Use the table shown to find lim
x→ 6
f(x).
(ii) Use the graph of f(x) = x(24- 2x) shown to find lim
x→ 6
f(x). Do you get the same limit as you did in part
(i)? What information about the limit is shown by the graph that might not be obvious from the table?
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[-5, 15, 5] by [0, 80, 10]
A) (i) 72
(ii) 72; Yes, the same limit is obtained. The graph shows that the limit is the maximum area.
B) (i) 80
(ii) 80; Yes, the same limit is obtained. The graph shows that the limit is the maximum area.
C) (i) 72
(ii) 80; No, the same limit is not obtained. The graph shows that the limit is the maximum amount of
material that can be used.
D) (i) 80
(ii) 72; No, the same limit is not obtained. The graph shows that the limit is the maximum amount of
material that can be used.
11.2 Finding Limits Using Properties of Limits
1 Find Limits of Constant Functions and the Identity Function
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit
properties can be applied.
1) lim
x→10
-6
A) -6 B) 10 C) 6 D) 0
2) lim
x→-8
x
A) -8 B) 8 C) 0 D) 1
2 Find Limits Using Properties of Limits
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit
properties can be applied.
1) lim
x→-7
15x3
A) -5145 B) -7 C) -343 D) 15
2) lim
x→0
(x2 - 5)
A) -5 B) 5 C) 0 D) does not exist
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