Calculus 2nd edition briggs test bank

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the average velocity of the function over the given interval.
1) y = x2 + 2x, [4, 8]
A) 10

1)

B) 14

C) 7

D) 20

B) - 688

321
C)
4

321
D) 4

2) y = 7x3 + 8x2 - 1, [-8, -4]
A) 688

2)

3) y =

2x, [2, 8]
1

A)
3

4) y =

3)
C) -

B) 2

3
10

D) 7

3
, [4, 7]
x-2

A)

4)

1
3

C) -

B) 7

5) y = 4x2 , 0,

3
10

D) 2

7
4

5)

A) 2

1
3

D) -

B) 7

C)

B) -34

1
C) 6

3
10

6) y = -3x2 - x, [5, 6]
A) -2

7) h(t) = sin (3t), 0,
A)

π
6

6
π

7)
B)

8) g(t) = 3 + tan t, A) -

6)

8
5

1
D)
2

3
π

C)

π
6

D) -

π π
,
4 4

6
π

8)
B)

4
π

C) -

1

4
π

D) 0

Use the table to find the instantaneous velocity of y at the specified value of x.
9) x = 1.
x y
0 0
0.2 0.02
0.4 0.08
0.6 0.18
0.8 0.32
1.0 0.5
1.2 0.72
1.4 0.98
A) 2

B) 0.5

C) 1.5

9)

D) 1

10) x = 1.
x y
0 0
0.2 0.01
0.4 0.04
0.6 0.09
0.8 0.16
1.0 0.25
1.2 0.36

1.4 0.49
A) 1

10)

B) 1.5

C) 2

D) 0.5

11) x = 1.
x y
0 0
0.2 0.12
0.4 0.48
0.6 1.08
0.8 1.92
1.0 3
1.2 4.32
1.4 5.88
A) 6

11)

B) 8

C) 4

2

D) 2


12) x = 2.

12)

x y
0 10
0.5 38
1.0 58
1.5 70
2.0 74
2.5 70
3.0 58
3.5 38
4.0 10
A) 4

B) 0

D) -8

C) 8

13) x = 1.

13)

x
y
0.900 -0.05263
0.990 -0.00503
0.999 -0.0005
1.000 0.0000
1.001 0.0005
1.010 0.00498
1.100 0.04762
A) 0.5

B) -0.5

C) 1

D) 0

For the given position function, make a table of average velocities and make a conjecture about the instantaneous
velocity at the indicated time.
14) s(t) = t2 + 8t - 2 at t = 2
14)
t
s(t)

1.9

1.99

1.999

2.001

2.01

2.1

A)
t
1.9
1.99
1.999 2.001 2.01
2.1
; instantaneous velocity is 18.0
s(t) 16.810 17.880 17.988 18.012 18.120 19.210
B)
t
1.9
1.99
1.999 2.001 2.01
2.1
; instantaneous velocity is 17.70
s(t) 16.692 17.592 17.689 17.710 17.808 18.789
C)
t 1.9
1.99 1.999 2.001 2.01 2.1
; instantaneous velocity is ∞
s(t) 5.043 5.364 5.396 5.404 5.436 5.763
D)
t 1.9
1.99 1.999 2.001 2.01 2.1

; instantaneous velocity is 5.40
s(t) 5.043 5.364 5.396 5.404 5.436 5.763

3


15) s(t) = t2 - 5 at t = 0

15)

-0.1

t
s(t)

-0.01

-0.001

0.001

0.01

0.1

A)
t -0.1
s(t) -1.4970
-15.0

-0.01
-1.4999

-0.001
-1.5000

0.001
0.01
0.1
; instantaneous velocity is
-1.5000 -1.4999 -1.4970

t -0.1
s(t) -4.9900
-5.0

-0.01
-4.9999

-0.001
-5.0000

0.001
0.01
0.1
; instantaneous velocity is
-5.0000 -4.9999 -4.9900

t -0.1
s(t) -1.4970

-0.01
-1.4999

-0.001
-1.5000

0.001
0.01
0.1
; instantaneous velocity is ∞
-1.5000 -1.4999 -1.4970

t -0.1
s(t) -2.9910
-3.0

-0.01
-2.9999

-0.001
-3.0000

0.001
0.01
0.1
; instantaneous velocity is
-3.0000 -2.9999 -2.9910

B)

C)

D)

Find the slope of the curve for the given value of x.
16) y = x2 + 5x, x = 4
A) slope is 13

17) y = x2 + 11x - 15, x = 1
1
A) slope is
20

16)

1
B) slope is
20

4
C) slope is 25

B) slope is -39

4
C) slope is 25

D) slope is -39

17)
D) slope is 13

18) y = x3 - 7x, x = 1
A) slope is -3

B) slope is -4

C) slope is 3

D) slope is 1

19) y = x3 - 2x2 + 4, x = 3
A) slope is 1

B) slope is 0

C) slope is -15

D) slope is 15

20) y = -4 - x3, x = 1
A) slope is 0

B) slope is -1

C) slope is -3

D) slope is 3

18)

19)

20)

Solve the problem.
21) Given lim f(x) = Ll, lim f(x) = Lr, and Ll ≠ Lr, which of the following statements is true?
x→0 x→0 +
I.

lim f(x) = Ll
x→0

II.

lim f(x) = Lr
x→0

III. lim f(x) does not exist.
x→0
A) none

B) II

C) III

4

D) I

21)


22) Given

lim f(x) = Ll, lim f(x) = Lr , and Ll = Lr, which of the following statements is false?
x→0 x→0 +

I.

lim f(x) = Ll
x→0

II.

lim f(x) = Lr
x→0

22)

III. lim f(x) does not exist.
x→0
A) I

B) II

C) III

D) none

23) If lim f(x) = L, which of the following expressions are true?
x→0
I.

lim f(x) does not exist.
x→0 -

II.

lim f(x) does not exist.
x→0 +

III.

lim f(x) = L
x→0 -

IV.

lim f(x) = L
x→0 +

A) I and II only

B) III and IV only

C) II and III only

23)

D) I and IV only

24) What conditions, when present, are sufficient to conclude that a function f(x) has a limit as x
approaches some value of a?
A) f(a) exists, the limit of f(x) as x→a from the left exists, and the limit of f(x) as x→a from the
right exists.
B) The limit of f(x) as x→a from the left exists, the limit of f(x) as x→a from the right exists, and
at least one of these limits is the same as f(a).
C) The limit of f(x) as x→a from the left exists, the limit of f(x) as x→a from the right exists, and
these two limits are the same.
D) Either the limit of f(x) as x→a from the left exists or the limit of f(x) as x→a from the right
exists

5

24)


Use the graph to evaluate the limit.
25) lim f(x)
x→-1

25)

y

1

-6 -5 -4 -3 -2 -1

1

2

3

4

5

B) -

1
2

6 x

-1

A) -1

C)

1
2

D) ∞

26) lim f(x)

x→0

26)
y
4
3
2
1

-4

-3

-2

-1

1

2

3

4 x

-1
-2
-3
-4

A) does not exist

B) -2

C) 0

6

D) 2


27) lim f(x)
x→0

27)
6

y

5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1

1

2

3

4

5

6 x

-2
-3
-4
-5
-6

A) does not exist

B) 3

C) 0

D) -3

28) lim f(x)
x→0

28)

12

y

10
8
6
4
2
-2

-1

1

2

3

4

5

x

-2
-4

A) -1

B) 6

C) does not exist

7

D) 0


29) lim f(x)
x→0

29)

y
4
3
2
1
-4

-3

-2

-1

1

2

3

4 x

-1
-2
-3
-4

A) 1

C) ∞

B) does not exist

D) -1

30) lim f(x)
x→0

30)

y
4
3
2
1
-4

-3

-2

-1

1

2

3

4 x

-1
-2
-3
-4

A) -1

B) does not exist

C) 1

8

D) ∞


31) lim f(x)
x→0

31)
y
4
3
2
1

-4

-3

-2

-1

1

2

3

4

x

-1
-2
-3
-4

A) does not exist

B) 0

C) 2

D) -2

32) lim f(x)
x→0

32)
y
4
3
2
1

-4

-3

-2

-1

1

2

3

4

x

-1
-2
-3
-4

A) -2

B) does not exist

C) 1

9

D) 0


33) lim f(x)
x→0

33)
y
4
3

2
1

-4

-3

-2

-1

1

2

3

x

4

-1
-2
-3
-4

A) -1
34) Find

B) does not exist

C) 2

D) -2

lim f(x) and
lim
f(x)
x→(-1)x→(-1)+

34)

y
2

-4

-2

2

4

x

-2

-4

-6

A) -2; -7

B) -5; -2

C) -7; -5

10

D) -7; -2


Use the table of values of f to estimate the limit.
35) Let f(x) = x2 + 8x - 2, find lim f(x).
x→2
x
f(x)

1.9

1.99

1.999

35)

2.001

2.01

2.1

A)
x
1.9
1.99
1.999 2.001 2.01
2.1
; limit = 17.70
f(x) 16.692 17.592 17.689 17.710 17.808 18.789
B)
x 1.9
1.99 1.999 2.001 2.01 2.1
; limit = ∞
f(x) 5.043 5.364 5.396 5.404 5.436 5.763
C)
x
1.9
1.99
1.999 2.001 2.01
2.1
; limit = 18.0
f(x) 16.810 17.880 17.988 18.012 18.120 19.210
D)
x 1.9
1.99 1.999 2.001 2.01 2.1
; limit = 5.40
f(x) 5.043 5.364 5.396 5.404 5.436 5.763
36) Let f(x) =

x
f(x)

x-4
, find lim f(x).
x-2
x→4
3.9

3.99

3.999

36)

4.001

4.01

4.1

A)
x 3.9
3.99
3.999
4.001
4.01
4.1
; limit = 1.20
f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745

B)
x 3.9
3.99
3.999
4.001
4.01
4.1
; limit = ∞
f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745
C)
x 3.9
3.99
3.999
4.001
4.01
4.1
; limit = 4.0
f(x) 3.97484 3.99750 3.99975 4.00025 4.00250 4.02485
D)
x 3.9
3.99
3.999
4.001
4.01
4.1
; limit = 5.10
f(x) 5.07736 5.09775 5.09978 5.10022 5.10225 5.12236

11

37) Let f(x) = x2 - 5, find lim f(x).
x→0
-0.1

x
f(x)

-0.01

37)

-0.001

0.001

0.01

0.1

A)
x -0.1
f(x) -2.9910

-0.01
-2.9999

-0.001
-3.0000

0.001
0.01
0.1
; limit = -3.0
-3.0000 -2.9999 -2.9910

x -0.1
f(x) -4.9900

-0.01
-4.9999

-0.001
-5.0000

0.001
0.01
0.1
; limit = -5.0
-5.0000 -4.9999 -4.9900

x -0.1
f(x) -1.4970

-0.01
-1.4999

-0.001
-1.5000

0.001
0.01
0.1
; limit = ∞
-1.5000 -1.4999 -1.4970

x -0.1
f(x) -1.4970

-0.01
-1.4999

-0.001
-1.5000

0.001
0.01
0.1
; limit = -15.0
-1.5000 -1.4999 -1.4970

B)

C)

D)

38) Let f(x) =

x

f(x)

x-5
x2 - 8x + 15

, find lim f(x).
x→5

4.9

4.99

4.999

38)

5.001

5.01

5.1

A)
x
4.9
4.99
4.999 5.001
5.01
5.1 ; limit = 0.5
f(x) 0.5263 0.5025 0.5003 0.4998 0.4975 0.4762

B)
x
4.9
4.99
4.999
5.001
5.01
5.1
; limit = -0.5
f(x) -0.5263 -0.5025 -0.5003 -0.4998 -0.4975 -0.4762
C)
x
4.9
4.99
4.999 5.001
5.01
5.1 ; limit = 0.4
f(x) 0.4263 0.4025 0.4003 0.3998 0.3975 0.3762
D)
x
4.9
4.99
4.999 5.001
5.01
5.1 ; limit = 0.6
f(x) 0.6263 0.6025 0.6003 0.5998 0.5975 0.5762

12

39) Let f(x) =

x
f(x)

x2 - 3x + 2
, find lim f(x).
x2 + 3x - 10
x→2
1.9

1.99

39)

1.999

2.001

2.01

2.1

A)
x
1.9
1.99
1.999 2.001
2.01
2.1 ; limit = 0.0429

f(x) 0.0304 0.0416 0.0427 0.0430 0.0441 0.0549
B)
x
1.9
1.99
1.999
2.001
2.01
2.1
; limit = -1
f(x) -1.0690 -1.0067 -1.0007 -0.9993 -0.9934 -0.9355
C)
x
1.9
1.99
1.999 2.001
2.01
2.1 ; limit = 0.2429
f(x) 0.2304 0.2416 0.2427 0.2430 0.2441 0.2549
D)
x
1.9
1.99
1.999 2.001
2.01
2.1 ; limit = 0.1429
f(x) 0.1304 0.1416 0.1427 0.1430 0.1441 0.1549
40) Let f(x) =

x

f(x)

sin(2x)
, find lim f(x).
x
x→0

-0.1

-0.01
1.99986667

-0.001

40)

0.001

0.01
1.99986667

A) limit = 1.5
C) limit = 2
41) Let f(θ) =

0.1

B) limit = 0
D) limit does not exist

cos (6θ)
, find lim f(θ).
θ
θ→0

x
-0.1
f(θ) -8.2533561

-0.01

-0.001

41)

0.001

0.01

A) limit = 6
C) limit = 8.2533561

0.1
8.2533561

B) limit = 0
D) limit does not exist

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.

42) It can be shown that the inequalities 1 -

x2
x sin(x)
<
< 1 hold for all values of x close
6
2 - 2 cos(x)

to zero. What, if anything, does this tell you about

x sin(x)
? Explain.
2 - 2 cos(x)

13

42)


MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
43) Write the formal notation for the principle "the limit of a quotient is the quotient of the limits" and
include a statement of any restrictions on the principle.
lim g(x)
x→a
g(x)
M
A) If lim g(x) = M and lim f(x) = L, then lim
=
= , provided that

f(x)
lim
f(x)
L
x→a
x→a
x→a
x→a

43)

L ≠ 0.
g(x) g(a)
B) lim
, provided that f(a) ≠ 0.
=
f(a)
x→a f(x)
g(x) g(a)
C) lim
.
=
f(a)
x→a f(x)
lim g(x)
x→a
g(x)
M
D) If lim g(x) = M and lim f(x) = L, then lim
=

= , provided that
f(x)
lim
f(x)
L
x→a
x→a
x→a
x→a
f(a) ≠ 0.
44) Provide a short sentence that summarizes the general limit principle given by the formal notation
lim [f(x) ± g(x)] = lim f(x) ± lim g(x) = L ± M, given that lim f(x) = L and lim g(x) = M.
x→a
x→a
x→a
x→a
x→a

44)

A) The sum or the difference of two functions is the sum of two limits.
B) The limit of a sum or a difference is the sum or the difference of the functions.
C) The sum or the difference of two functions is continuous.
D) The limit of a sum or a difference is the sum or the difference of the limits.
45) The statement "the limit of a constant times a function is the constant times the limit" follows from
a combination of two fundamental limit principles. What are they?
A) The limit of a product is the product of the limits, and a constant is continuous.
B) The limit of a product is the product of the limits, and the limit of a quotient is the quotient of
the limits.
C) The limit of a function is a constant times a limit, and the limit of a constant is the constant.

D) The limit of a constant is the constant, and the limit of a product is the product of the limits.
Find the limit.
46) lim
x→20

10

A) 10

46)
B)

C) 2 5

10

D) 20

47) lim (6x - 4)
x→1
A) -2

47)
C) -10

B) 2

D) 10

48) lim (12 - 10x)

x→7
A) 82

45)

48)
B) -82

C) -58

14

D) 58


Give an appropriate answer.
49) Let lim f(x) = -8 and lim g(x) = -5. Find lim [f(x) - g(x)].
x → -4
x → -4
x → -4
B) -4

A) -3

C) -8

49)
D) -13

50) Let lim f(x) = -10 and lim g(x) = 8. Find lim [f(x) ∙ g(x)].

x→2
x→2
x→2
A) -2

51) Let

D) 2

f(x)
lim f(x) = -3 and lim g(x) = 6. Find lim
.
g(x)
x → -5
x → -5
x → -5

A) -

52) Let

C) -80

B) 8

50)

1
2

lim f(x) = 64. Find lim
x → 10
x → 10

A) 8

C) -5

B) - 2

51)
D) -9

f(x).

52)

B) 2.8284

C) 64

D) 10

53) Let lim f(x) = -2 and lim g(x) = -7. Find lim [f(x) + g(x)]2 .
x→5
x→5
x→5
A) 81

B) -9

54) Let lim f(x) = 243. Find lim
x→8
x→8
A) 3

5

C) 53

54)

B) 243

22
5

D) 5

f(x).

55) Let lim f(x) = -9 and lim g(x) = 1. Find lim
x→ 5
x→ 5
x→ 5
A)

53)

C) 8

D) 5

-4f(x) - 8g(x)
.
9 + g(x)

B) 5

C) - 4

55)
D)

14
5

Find the limit.
56) lim (x3 + 5x2 - 7x + 1)
x→2
A) 15
57)

B) 29

C) does not exist

D) 0

lim (3x5 - 3x4 + 4x3 + x2 - 5)

x→-2
A) -177

58)

56)

57)

B) -113

C) -81

D) -33

x
lim
3x
+2
x→-1
A) -

1
5

58)
B) 1

C) does not exist

15

D) 0


59) lim
x→0

x3 - 6x + 8
x-2

A) Does not exist

59)
C) -4

B) 0

D) 4

3x2 + 7x - 2
60) lim
x→1 3x2 - 4x - 2

60)
B) -

A) 0

7

4

C) Does not exist

D) -

8
3

61) lim (x + 3)2 (x - 1)3
x→2
A) 1
62) lim
x→2

B) 27

C) 675

D) 25

x2 + 2x + 1

A) 3
63) lim
x→9

61)

62)

C) ±3

B) 9

D) does not exist

4x + 65

A) -101

64) lim
h→0

A) 1/2

B)

C) 101

101

D) - 101

2
3h + 4 + 2

A) 1

65) lim
x→0

63)

64)
B) 2

C) 1/2

D) Does not exist

1+x-1
x

65)
B) Does not exist

C) 1/4

D) 0

Determine the limit by sketching an appropriate graph.
for x < 2
66) lim f(x), where f(x) = -2x - 7
4x
6
for x ≥ 2
x → 2A) -5
67)

68)

B) -6

lim f(x), where f(x) = -3x - 4
4x - 3
x → 4+
A) 13

C) -11

D) 2

for x < 4
for x ≥ 4

67)

B) -2

2
lim f(x), where f(x) = x + 3
0
x → -4+
A) 16

66)

C) -16

D) -3

for x ≠ -4
for x = -4

68)

B) 13

C) 0

16

D) 19


69)

lim f(x), where f(x) =
x → 4A) 0

70)

lim f(x), where f(x) =
x → -7+
A) -21

4 - x2

0≤x<2
2

2≤x<4
4
x=4
B) Does not exist
3x
3
0
B) -0

69)
C) 4

-7 ≤ x < 0, or 0 < x ≤ 3
x=0
x < -7 or x > 3
C) Does not exist

D) 2

70)
D) 5

Find the limit, if it exists.
x3 + 12x2 - 5x
71) lim
5x
x→0
A) Does not exist

71)

B) 0

C) 5

D) -1

x4 - 1
72) lim
x→1 x - 1
A) 0

73)

B) Does not exist

lim
x→6

lim
x→6
A)

77)

73)
B) 1

C) 14

D) 7

74)
B) Does not exist

C) 224

D) 16

x2 + 4x - 60
x-6

A) 16

76)

D) 2

x2 + 16x + 63
x+7

lim
x → -7
A) 2

75)

C) 4

x2 - 49
lim

x→7 x-7
A) Does not exist

74)

72)

75)
B) Does not exist

C) 0

D) 4

x2 + 4x - 60
x2 - 36

4
3

76)
B) -

1
3

C) 0

D) Does not exist

x2 - 25
lim
x → 5 x2 - 7x + 10
A) 0

77)
B)

10
3

C) Does not exist

D)

5
3

x2 + 2x - 3
78) lim
x→1 x2 - 4x + 3
A) - 2

78)
B) 1

C) 2

17

D) Does not exist


79)

lim
h→0

(x + h)3 - x3
h

A) 3x2

80)

79)
B) 3x2 + 3xh + h 2

C) 0

D) Does not exist

6-x
6-x

lim
x→6

80)

A) 0

B) 1

D) -1

C) Does not exist

Provide an appropriate response.
81) It can be shown that the inequalities -x ≤ x cos

1
≤ x hold for all values of x ≥ 0.
x

81)

1
Find lim x cos
if it exists.
x
x→0
A) 0.0007

82) The inequality 1Find lim
x→0

B) does not exist

C) 0

D) 1

x2 sin x
<
< 1 holds when x is measured in radians and x < 1.
2
x

sin x
if it exists.
x

A) 0

B) 1

C) 0.0007

D) does not exist

83) If x3 ≤ f(x) ≤ x for x in [-1,1], find lim f(x) if it exists.
x→0
B) -1

A) 0

83)
C) 1

D) does not exist

Compute the values of f(x) and use them to determine the indicated limit.
84) If f(x) = x2 + 8x - 2, find lim f(x).
x→2
x
f(x)

1.9

82)

1.99

1.999

2.001

2.01

84)

2.1

A)
x
1.9
1.99
1.999 2.001 2.01
2.1

; limit = 18.0
f(x) 16.810 17.880 17.988 18.012 18.120 19.210
B)
x
1.9
1.99
1.999 2.001 2.01
2.1
; limit = 17.70
f(x) 16.692 17.592 17.689 17.710 17.808 18.789
C)
x 1.9
1.99 1.999 2.001 2.01 2.1
; limit = 5.40
f(x) 5.043 5.364 5.396 5.404 5.436 5.763
D)
x 1.9
1.99 1.999 2.001 2.01 2.1
; limit = ∞
f(x) 5.043 5.364 5.396 5.404 5.436 5.763

18


85) If f(x) =

x
f(x)

x4 - 1

, find lim f(x).
x-1
x→1
0.9

0.99

85)

0.999

1.001

1.01

1.1

A)
x 0.9
0.99 0.999 1.001 1.01 1.1
; limit = 4.0
f(x) 3.439 3.940 3.994 4.006 4.060 4.641
B)
x 0.9
0.99 0.999 1.001 1.01 1.1
; limit = ∞
f(x) 1.032 1.182 1.198 1.201 1.218 1.392
C)
x 0.9
0.99 0.999 1.001 1.01 1.1

; limit = 5.10
f(x) 4.595 5.046 5.095 5.105 5.154 5.677
D)
x 0.9
0.99 0.999 1.001 1.01 1.1
; limit = 1.210
f(x) 1.032 1.182 1.198 1.201 1.218 1.392

86) If f(x) =

x
f(x)

x3 - 6x + 8
, find lim f(x).
x-2
x→0
-0.1

-0.01

-0.001

86)

0.001

0.01

0.1

A)
x -0.1
0.001
0.01
0.1
-0.01
-0.001
; limit = ∞
f(x) -1.22843 -1.20298 -1.20030 -1.19970 -1.19699 -1.16858
B)
x -0.1
0.001
0.01
0.1
-0.01
-0.001
; limit = -1.20
f(x) -1.22843 -1.20298 -1.20030 -1.19970 -1.19699 -1.16858
C)
x -0.1
0.001
0.01
0.1
-0.01
-0.001
; limit = -4.0
f(x) -4.09476 -4.00995 -4.00100 -3.99900 -3.98995 -3.89526
D)
x -0.1

0.001
0.01
0.1
-0.01
-0.001
; limit = -2.10
f(x) -2.18529 -2.10895 -2.10090 -2.99910 -2.09096 -2.00574

19


87) If f(x) =

x
f(x)

x-4
, find lim f(x).
x-2
x→4
3.9

3.99

87)

3.999

4.001

4.01

4.1

A)
x 3.9
3.99
3.999
4.001
4.01
4.1
; limit = 1.20
f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745
B)
x 3.9
3.99
3.999
4.001
4.01
4.1
; limit = 4.0
f(x) 3.97484 3.99750 3.99975 4.00025 4.00250 4.02485
C)
x 3.9
3.99
3.999
4.001
4.01
4.1
; limit = ∞

f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745
D)
x 3.9
3.99
3.999
4.001
4.01
4.1
; limit = 5.10
f(x) 5.07736 5.09775 5.09978 5.10022 5.10225 5.12236
88) If f(x) = x2 - 5, find

x
f(x)

-0.1

lim f(x).
x→0
-0.01

88)

-0.001

0.001

0.01

0.1

A)
x -0.1
f(x) -2.9910

-0.01
-2.9999

-0.001
-3.0000

0.001
0.01
0.1
; limit = -3.0
-3.0000 -2.9999 -2.9910

x -0.1
f(x) -1.4970

-0.01
-1.4999

-0.001
-1.5000

0.001
0.01
0.1
; limit = -15.0

-1.5000 -1.4999 -1.4970

x -0.1
f(x) -1.4970

-0.01
-1.4999

-0.001
-1.5000

0.001
0.01
0.1
; limit = ∞
-1.5000 -1.4999 -1.4970

x -0.1
f(x) -4.9900

-0.01
-4.9999

-0.001
-5.0000

0.001
0.01
0.1
; limit = -5.0

-5.0000 -4.9999 -4.9900

B)

C)

D)

20


89) If f(x) =

x
f(x)

x+1
, find lim f(x).
x+1
x→1
0.9

0.99

89)

0.999

1.001

1.01

1.1

A)
x 0.9
0.99
0.999
1.001
1.01
1.1
; limit = 2.13640
f(x) 2.15293 2.13799 2.13656 2.13624 2.13481 2.12106
B)
x 0.9
0.99
0.999
1.001
1.01
1.1
; limit = 0.7071
f(x) 0.72548 0.70888 0.70728 0.70693 0.70535 0.69007
C)
x 0.9
0.99
0.999
1.001
1.01
1.1
; limit = ∞

f(x) 0.21764 0.21266 0.21219 0.21208 0.21160 0.20702
D)
x 0.9
0.99
0.999
1.001
1.01
1.1
; limit = 0.21213
f(x) 0.21764 0.21266 0.21219 0.21208 0.21160 0.20702
90) If f(x) =

x
f(x)

x - 2, find

3.9

lim f(x).
x→4

3.99

3.999

90)

4.001

4.01

4.1

A)
x
3.9
3.99
3.999
4.001
4.01
4.1
; limit = 0
f(x) -0.02516 -0.00250 -0.00025 0.00025 0.00250 0.02485
B)
x 3.9
f(x) 3.9000

3.99
2.9000

3.999
1.9000

4.001
4.01
4.1
; limit = 1.95
2.0000 3.0000 4.0000

x 3.9
f(x) 3.9000

3.99
2.9000

3.999
1.9000

4.001
4.01
4.1
; limit = ∞
2.0000 3.0000 4.0000

C)

D)
x 3.9
3.99
3.999
4.001
4.01
4.1
; limit = 1.50
f(x) 1.47736 1.49775 1.49977 1.50022 1.50225 1.52236

21

For the function f whose graph is given, determine the limit.
91) Find lim f(x) and
lim f(x).
x→5 x→5 +

91)

y
8
6
4
2
1 2 3 4 5 6 7 8 9 10 11 x

-2 -1
-2
-4
-6
-8

B) ∞, -∞

A) -5, 5
92) Find

lim f(x) and
x→2 5

C) -∞, ∞

D) 5; 5

lim f(x).
x→2 +

92)

y

4
3
2
1
-5

-4

-3

-2

-1

1

2

3

4

5 x

-1
-2
-3
-4
-5

A) ∞; ∞

B) 2; -2

C) 0; 1

22

D) -∞; ∞


93) Find lim f(x).
x→3

93)
5

y

4
3

2
1
-5

-4

-3

-2

-1

1

2

3

4

5 x

-1
-2
-3
-4
-5

A) -∞

C) ∞

B) 3

D) does not exist

94) Find lim f(x).
x→-3

94)

6

y

5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1

1

2

3

4

5

6 x

-2
-3
-4
-5
-6

A) -∞

C) -3

B) 0

D) ∞

95) Find lim f(x).
x→0

95)
y
4

2

-4

-2

2

4

x

-2

-4

A) ∞

B) 0

C) 1

23

D) -∞


Find the limit.
96)

1
lim
x
x→-2 + 2

A) Does not exist

97)

D) -1
98)

C) -1

D) ∞

99)
C) ∞

B) 0

D) -∞

1
lim
2
x
- 16
x → 4+

100)
B) 0

C) 1

D) ∞

lim
tan x
x→(π/2)+

101)
B) ∞

C) 1

D) -∞

lim
sec x
x→(-π/2)-

102)
B) ∞

A) 0
103)

C) -∞

7
lim
2
x → -3- x - 9

A) 0
102)

97)

B) 0

A) -∞
101)

D) 1/2

1
lim
2
x → 3 - (x - 3)

A) -1

100)

C) -∞

B) 0

A) -∞

99)

B) ∞

1
lim
x
+3
x → -3A) ∞

98)

96)

C) -∞

D) 1

lim (1 + csc x)
x→0+
A) ∞

103)
B) 1

C) 0

D) Does not exist

104) lim (1 - cot x)
x→0
A) ∞

105)

lim
x → -2+

B) -∞

lim
x → 2+
A) -∞

C) 0

D) Does not exist

x2 - 7x + 10
x3 - 4x

A) 0

106)

104)

105)
B) -∞

C) Does not exist

D) ∞

x2 - 5x + 6
x3 - 9x

106)
B) ∞

C) Does not exist

24

D) 0


Find all vertical asymptotes of the given function.
3x
107) f(x) =
x+4
A) x = 4
108) f(x) =

107)

B) x = 3

C) x = -4

x+5
x2 - 64

108)

A) x = 64, x = -5
C) x = -8, x = 8
109) g(x) =

B) x = 0, x = 64
D) x = -8, x = 8, x = -5

x+5
x2 + 1

109)

A) x = -1, x = 1
C) x = -1, x = 1, x = -5
110) f(x) =

B) x = -1, x = -5
D) none

x + 11
x2 + 25x

110)

A) x = -5, x = 5
C) x = -25, x = -11
111) f(x) =

B) x = 0, x = -25
D) x = 0, x = -5, x = 5

x-1
3
x + 16x

111)

A) x = 0
C) x = 0, x = -16

112) R(x) =

B) x = 0, x = -4, x = 4
D) x = -4, x = 4

-3x2

112)

x2 + 4x - 21

A) x = -7, x = 3
C) x = - 21
113) R(x) =

B) x = -7, x = 3, x = -3
D) x = 7, x = -3

x-1

113)

x3 + 3x2 - 28x

A) x = -4, x = -30, x = 7
C) x = -7, x = 4
114) f(x) =

B) x = -4, x = 0, x = 7
D) x = -7, x = 0, x = 4

-2x(x + 2)

114)

2x2 - 5x - 7

A) x = -

115) f(x) =

D) none

2
,x=1
7

B) x = -

7
,x=1
2

C) x =

2
, x = -1
7

x-3
9x - x3

D) x =

7
, x = -1
2

115)

A) x = -3, x = 3
C) x = 0, x = -3

B) x = 0, x = -3, x = 3
D) x = 0, x = 3

25