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# Chapter 1: Limits and Their Properties

1. Decide whether the following problem can be solved using precalculus, or whether
calculus is required. If the problem can be solved using precalculus, solve it. If the
problem seems to require calculus, use a graphical or numerical approach to estimate the
solution.
Find the distance traveled in 11 seconds by an object moving with a velocity of
v(t ) = 6 + 3cos t .
A) precalculus , 63.0000
D) precalculus , 66.0133
B) calculus , 66.0133
E) precalculus , 67.3633
C) calculus , 63.0000
2. Decide whether the following problem can be solved using precalculus, or whether
calculus is required. If the problem can be solved using precalculus, solve it. If the
problem seems to require calculus, use a graphical or numerical approach to estimate the
solution.
Find the distance traveled in 12 seconds by an object traveling at a constant velocity of
12 feet per second.
A) precalculus , 288
D) calculus , 288
B) precalculus , 144
E) calculus , 164
C) calculus , 144

17

## Chapter 1: Limits and Their Properties

3. Decide whether the following problem can be solved using precalculus, or whether
calculus is required. If the problem can be solved using precalculus, solve it. If the
problem seems to require calculus, use a graphical or numerical approach to estimate the
solution.
A cyclist is riding on a path whose elevation is modeled by the function
f ( x) = 0.09 (18 x x 2 ) where x and f ( x) are measured in miles. Find the rate of change
of elevation when x = 4.5.

A) precalculus , 0.21
B) calculus , 0.09
C) precalculus , 0.21

D) precalculus , 0.09
E) calculus , 0.81

4. Decide whether the following problem can be solved using precalculus, or whether
calculus is required. If the problem can be solved using precalculus, solve it. If the
problem seems to require calculus, use a graphical or numerical approach to estimate the
solution.
A cyclist is riding on a path whose elevation is modeled by the function f ( x) = 0.16 x
where x and f ( x) are measured in miles. Find the rate of change of elevation when
x = 4.0.

A) calculus , 1.28
B) precalculus , 1.28
C) calculus , 0.16

18

D) precalculus , 0.16
E) precalculus , 0.41

## Chapter 1: Limits and Their Properties

5. Decide whether the following problem can be solved using precalculus, or whether
calculus is required. If the problem can be solved using precalculus, solve it. If the
problem seems to require calculus, use a graphical or numerical approach to estimate the
solution.
Find the area of the shaded region.

A) precalculus , 22
B) calculus , 22
C) precalculus , 15

D) calculus , 15
E) precalculus , 18

6. Decide whether the following problem can be solved using precalculus, or whether
calculus is required. If the problem can be solved using precalculus, solve it. If the
problem seems to require calculus, use a graphical or numerical approach to estimate the
solution.
Find the area of the shaded region bounded by the triangle with vertices (0,0), (3,4),
(7,0).

A) precalculus , 14.0
B) precalculus , 28.0
C) calculus , 42.0

D) calculus , 28.0
E) precalculus , 42.0

19

## 7. Consider the function f ( x) = x and the point P(100,10) on the graph of f.

Consider the secant lines passing through P(100,10) and Q(x,f(x)) for x values of 97, 99,
and 101. Find the slope of each secant line to four decimal places.
(Think about how you could use your results to estimate the slope of the tangent line of f
at P(100,10), and how to improve your approximation of the slope.)
A) 0.0504 , 0.0501 , 0.0499
D) 0.0252 , 0.0251 , 0.0249
B) 0.0252 , 0.0251 , 0.0249
E) 0.0504 , 0.0501 , 0.0499
C) 0.0504 , 0.0501 , 0.0499
8. Use the rectangles in each graph to approximate the area of the region bounded by the
following.
y = 9/x, y = 0, x = 1, and x = 9.

A) 30.1714 , 24.4607
B) 30.1714 , 48.9214
C) 60.3429 , 24.4607

D) 60.3429 , 48.9214
E) 45.2571 , 36.6911

9. Use the rectangles in the following graph to approximate the area of the region bounded
by y = sin x , y = 0, x = 0, and x = .

20

## Chapter 1: Limits and Their Properties

10. Use the rectangles in the following graph to approximate the area of the region bounded
by y = cos x , y = 0, x =

, and x =

## A) 3.9082 B) 2.6055 C) 0.9770 D) 1.4656 E) 1.9541

11. Complete the table and use the result to estimate the limit.
x5
lim 2
x 5 x 20 x + 75
4.9
4.99
4.999
5.001
5.01
5.1
x
f(x)
A) 0.100000 B) 0.400000 C) 0.275000 D) 0.525000 E) 0.475000

12. Complete the table and use the result to estimate the limit.
x + 12 5
lim
x 7
x+7
7.1
7.01
7.001
6.999
6.99
6.9
x
f(x)
A) 0.223607 B) 0.348607 C) 0.223607 D) 0.390273 E) 0.473607

13. Complete the table and use the result to estimate the limit.
1
1

lim x + 3 8
x 5
x5
4.9
4.99
4.999
5.001
5.01
5.1
x
f(x)
A) 0.114375 B) 0.094375 C) 0.145625 D) 0.015625 E) 0.125625

21

## Chapter 1: Limits and Their Properties

14. Complete the table and use the result to estimate the limit.
sin 2 x
lim 2
x 0
x
0.1
0.01
0.001
x
f(x)
A) 1 B) 0.5 C) 1 D) 0.5 E) 0

0.001

0.01

0.1

0.01

0.1

15. Complete the table and use the result to estimate the limit.
cos ( 4 x ) 1
lim
x 0
4 x
0.1
0.01
0.001
x
f(x)
A) 1 B) 0 C) 1 D) 2 E) 2

0.001

16. Determine the following limit. (Hint: Use the graph of the function.)
lim ( 5 x )
x 2

22

## Chapter 1: Limits and Their Properties

17. Let
3 x, x 1
.
f ( x) =
x =1
0
Determine the following limit. (Hint: Use the graph of the function.)

lim f ( x)
x 1

## A) 2 B) 0 C) 3 D) 4 E) Does not exist

18. Determine the following limit. (Hint: Use the graph of the function.)
lim ( x 2 + 3)
x 2

23

## Chapter 1: Limits and Their Properties

19. Let
x 2 + 4, x 1
.
f ( x) =
=
x
1,
1

Determine the following limit. (Hint: Use the graph of the function.)

lim f ( x)
x 1

## A) 5 B) 1 C) 4 D) 16 E) Does not exist.

20. Determine the following limit. (Hint: Use the graph of the function.)
lim
x 3

1
x 3

## A) 0 B) Does not exist C) 3 D) 3 E) 6

21. Let f ( x) = 3x 5 and g ( x) = x 2 . Find the limits:
(a) lim f ( x)
x 4

A) 12 , 4 , 23
B) 4 , 2 , 16
C) 8 , 2 , 23

24

(b) lim g ( x)
x 2

x 4

D) 12 , 2 , 25
E) 7 , 4 , 289

(a) lim f ( x)

(b) lim g ( x)

x 5

## (c) lim g ( f ( x))

x 2

x 3

A) 2 , 3 , 26 B) 5 , 2 , 27 C) 26 , 6 , 30 D) 5 , 2 , 1 E) 6 , 2 , 8
23. Let f ( x) = 5 + x 2 and g ( x) = x + 3 . Find the limits:

(a) lim f ( x)

(b) lim g ( x)

x 2

x 5

A) 9, 8, 33

x 5

B) 2, 5, 5

C) 6,5, 30

D) 4, 3, 8

E) 6, 5, 33

(a) lim f ( x)

(b) lim g ( x)

x 2

## (c) lim g ( f ( x))

x 2

x 5

D) 24, 3 2, 3 134
E) None of the above

A) 24, 3 2, 3 134
B) 28, 3 2, 3 134
C) 28, 3 2, 3 134
25. Find the limit:
lim sin x
x

3
4

A)

21/ 2
2

B)

21/ 2
2

C)

2 1/ 2
4

C)

31/ 2
2

D) 0 E)

2 1/ 2
4

x
lim cos

x 5
6
31/ 2
31/ 2
A)
B)
4
2

31/ 2
4

5x
lim tan
x
6
1/ 2
A) 3
B) 31/ 2

C) 61/ 2

25

## Chapter 1: Limits and Their Properties

28. Suppose that lim f ( x) = 7 and lim g ( x) = 4 . Find the following limit:
x c

lim f ( x)
x c

A) 11

x c

B) 3 C) 0 D) 28 E) 2401

g ( x)

29. Suppose that lim f ( x) = 8 and lim g ( x) = 7 . Find the following limit:
x c

lim [ f ( x) + g ( x) ]

x c

x c

A) 56 B) 15 C) 0 D) 1 E) 7
30. Suppose that lim f ( x) = 9 and lim g ( x) = 14 . Find the following limit:
x c

lim [ f ( x) g ( x) ]

x c

x c

A) 0 B) 5 C) 23 D) 126 E) 9
31. Suppose that lim f ( x) = 7 and lim g ( x) = 13 . Find the following limit:
lim [ 9 g( x) ]

x c

x c

x c

A) 117 B) 63 C) 9 D) 13 E) 7
32. Suppose that lim f ( x) = 14 and lim g ( x) = 15 . Find the following limit:
x c

lim [ f ( x) g ( x) ]

x c

x c

A) 14 B) 29 C) 1 D) 210 E) 15
33. Suppose that lim f ( x) = 6 and lim g ( x) = 9 . Find the following limit:
x c

f ( x)
g ( x)
3
2
A)
B) 54 C)
2
3

x c

lim
x c

## D) Does not exist. E) 54

34. Find the following limit (if it exists). Write a simpler function that agrees with the given
function at all but one point.
x3 + 512
x 8
x +8
A) 192 , x 2 8 x + 64
B) 64 , x 2 + 8 x + 64
C) 64 , x 2 8 x 64
lim

26

## D) Limit does not exist.

E) 192 , x 2 8 x + 64

## Chapter 1: Limits and Their Properties

35. Find the following limit (if it exists). Write a simpler function that agrees with the given
function at all but one point.
12 x 2 179 x + 154
x 14
x 14
A) Does not exist.
B) 179 , 12 x + 11
C) 179 , 12 x 11
lim

D) 157 , 12 x + 11
E) 157 , 12 x 11

lim

x 8

x 8
x 2 64

A) 32 B)

1
16

C)

1
16

1
32

D) 8 E)

## 37. Find the limit (if it exists):

lim

( x + x )

( x + x ) 6 ( x 2 x 6 )

x 0

1
1
A) x 3 x 2 6 x
3
2

x
B) x3 x 2 6 x

C) 0 D) 2 x 1

E) x 2 x 6

## 38. Determine the limit (if it exists):

sin x (1 cos x )
x 0
2 x8
A) 0 B) 1 C) Does not exist. D) 2 E) 8
lim

## 39. Determine the limit (if it exists):

2 (1 cos x )
x 0
x
A) 0 B) 4 C) 8 D) Does not exist E) 4
lim

## 40. Determine the limit (if it exists):

sin 5 x
lim 4
x 0
x
A) B) 1 C) 0 D) Does not exist. E) 2

27

## Chapter 1: Limits and Their Properties

41. Use the graph as shown to determine the following limits, and discuss the continuity of
the function at x = 4 .
(i) lim+ f ( x)
x 4

(ii) lim f ( x)
x 4

(iii) lim f ( x)
x 4

A) 4 , 4, 4, Not continuous
B) 3 , 3 , 3 , Not continuous
C) 2 , 2 , 2 , Continuous

D) 3 , 3 , 3 , Continuous
E) 2 , 2 , 2 , Not continuous

42. Use the graph as shown to determine the following limits, and discuss the continuity of
the function at x = 3 .
(i) lim+ f ( x)
x 3

(ii) lim f ( x)
x 3

A) 1 , 1 , 1 , Not continuous
B) 1 , 1 , 1 , Continuous
C) 2 , 2 , 2 , Not continuous

28

(iii) lim f ( x)
x 3

D) 2 , 2 , 2 , Continuous
E) 3 , 3 , 3 , Continuous

## Chapter 1: Limits and Their Properties

43. Use the graph to determine the following limits, and discuss the continuity of the
function at x = 4 .
(i) lim+ f ( x)
x 4

(ii) lim f ( x)
x 4

(iii) lim f ( x)
x 4

## A) 2 , 0 , Does not exist , Not continuous

B) 0 , 2 , Does not exist , Not continuous
C) 0 , 2 , 0 , Continuous

## D) 4 , 0 , Does not exist , Not continuous

E) 2 , 2 , Does not exist , Not continuous

44. Find the limit (if it exists). Note that f ( x) = x represents the greatest integer function.
lim ( 4 x 7 )

x 8 +

## A) 39 B) 35 C) Does not exist D) 39 E) 35

45. Find the x-values (if any) at which the function f ( x) = 3 x 2 7 x 11 is not continuous.
Which of the discontinuitites are removable?
A) Continuous everywhere
D)
7
x = . Not removable.
6
E) both B and C
B) x = 11 . Removable
C)
7
x = . Removable.
6
46.

## Find the x-values (if any) at which the function f ( x) =

x
is not continuous.
x + 64
2

## Which of the discontinuitites are removable?

A) 8 and -8. Not removable.
D) Discontinuous everywhere
B) Continuous everywhere
E) None of the above
C) 8 and -8. Removable.

29

47.

## Find the x-values (if any) at which the function f ( x) =

x5
is not continuous.
x 6x + 5
2

## Which of the discontinuitites are removable?

A) No points of discontinuity.
B) x = 5 (Not removable), x = 1 (Removable)
C) x = 5 (Removable), x = 1 (Not removable)
D) No points of continuity.
E) x = 5 (Not removable), x = 1 (Not removable)
48. Find constants a and b such that the function
x 5
9,

f ( x) = ax + b, 5 < x < 1
9,
x 1

A) a = 3 , b = 0
B) a = 3 , b = 6
C) a = 3 , b = 6

D) a = 3 , b = 6
E) a = 3 , b = 6

## 49. Find the constant a such that the function

sin x
, x<0
4
f ( x) =
x
a + 15 x, x 0
is continuous on the entire real line.
A) 4 B) 4 C) 15 D) 15 E) 1
50.

x2
approaches or as x approaches 9 from the
x 2 81
left and from the right by completing the tables below.

Determine whether f ( x) =

x
f(x)

9.5

8.999
x
f(x)
A) lim f ( x) =
x 9

B)
C)

30

9.1

9.01

9.001

8.99

8.9

8.5

D)

lim f ( x) =

x 9+

lim f ( x) =

E) Both A and C

lim f ( x) =

F) Both B and D

x 9
x 9+

## Chapter 1: Limits and Their Properties

51.

52.

x 2 25
.
x 2 15 x + 50
A) x = 10 B) x = 10 C) x = 5 D) x = 5 E) x = 50

## Find the vertical asymptotes (if any) of the function f ( x) =

x 2 + 8 x + 12
.
x 3 + x 2 26 x + 24
A) x = 4 B) x = 1 C) x = 4 D) x = 1 E) A and B F) C and D
Find the vertical asymptotes (if any) of the function f ( x) =

53. Find the vertical asymptotes (if any) of the function f ( x) = tan(15 x ) .
A)
D) No vertical asymptotes
2k + 1
(k = 0, 1, 2, ...)
x=
30
B)
E)
2k + 1
2k
(k = 0, 1, 2, ...)
(k = 0, 1, 2, ...)
x=
x=
15
30
C)
2k
(k = 0, 1, 2, ...)
x=
15
54. Find the limit:
x8
x 6 x + 6
A) B)
lim+

C) 0 D) 1 E) 1

## 55. Find the limit:

lim

x 12

x 2 12 x
( x2 + 144 ) ( x 12 )

A) 24 B)

1
24

C) 24 D)

1
24

E) 12

1

lim x 4
x 0
x
A) 0 B)

C) 1 D) 1 E)

31

1. C
Section:
2. B
Section:
3. E
Section:
4. D
Section:
5. D
Section:
6. A
Section:
7. C
Section:
8. A
Section:
9. D
Section:
10. E
Section:
11. A
Section:
12. C
Section:
13. D
Section:
14. A
Section:
15. B
Section:
16. D
Section:
17. A
Section:
18. D
Section:
19. A
Section:
20. B
Section:
21. E
Section:
22. C
Section:

32

1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.3
1.3

23. A
Section:
24. D
Section:
25. B
Section:
26. C
Section:
27. A
Section:
28. E
Section:
29. D
Section:
30. C
Section:
31. A
Section:
32. D
Section:
33. C
Section:
34. A
Section:
35. E
Section:
36. C
Section:
37. D
Section:
38. C
Section:
39. A
Section:
40. C
Section:
41. E
Section:
42. A
Section:
43. B
Section:
44. D
Section:
45. A
Section:

1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.4
1.4
1.4
1.4
1.4

33

46. B
Section:
47. C
Section:
48. E
Section:
49. B
Section:
50. E
Section:
51. B
Section:
52. E
Section:
53. A
Section:
54. B
Section:
55. D
Section:
56. E
Section:

34

1.4
1.4
1.4
1.4
1.5
1.5
1.5
1.5
1.5
1.5
1.5
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