Chapter 1
Test Bank
Test Form A
Name
__________________________________________
Date
Chapter 1
Class
__________________________________________
Section _______________________
____________________________
1. Find lim 3x2 5.
x→2
(a) 41
(b) 17
(c) 11
(d) 0
(e) None of these
2. Given lim 2x 1 3. Find such that 2x 1 3 < 0.01 whenever 0 < x 2 < .
x→2
(a) 3
(b) 0.05
(d) 0.005
(e) None of these
3. Use the graph to find lim f x if f x x→1
(a) 2
31, x,
(c) 0.03
x1
.
x1
y
(b) 1
2
(c)
3
2
f
(d) The limit does not exist.
1
(e) None of these
x
−1
1
−1
x2 3x 2
.
x→1
x2 1
4. Find lim
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41
(a) 0
(b)
(d) The limit does not exist.
(e) None of these
(c) 1
5. Find lim x2 4.
x→3
(a) 1
(b) 5
(d) 5
(e) None of these
(c) 1
1
2
f x
6. If lim f x and lim gx , find lim
.
x→c
x→c
x→c gx
2
3
(a) 1
3
(d) 3
(b)
1
3
(e) None of these
(c) 3
4
2
3
Chapter 1
Limits and Their Properties
x2
.
x2 4
7. Find the limit: lim
x→2
1
4
(a) 0
(b)
(d) 1
(e) None of these
8. Find the limit: lim
x 4 2
x
x→0
1
4
(b)
(d) 1
(e) None of these
x→3
(c)
x3
.
x3
(a) 0
(b) 1
(d) The limit does not exist.
(e) None of these
10. Find the limit: lim sec
x→2
(c) 3
x
.
3
(a) 2
(d)
.
(a) 0
9. Find the limit: lim
(c)
(b)
1
2
2
3
(c) x→0
x
.
tan x
4
(a) 0
(b)
(d) The limit does not exist.
(e) None of these
(c) 1
12. Find the limit: lim 2x 5.
x→3
(a) 1
(b) 0
(d) The limit does not exist.
(e) None of these
13. Find the limit: lim
x→2
(d) 2
(e) None of these
11. Find the limit: lim
(a)
3
1
4
(c) 2i
1
.
x2
(b) (e) None of these
(c) 0
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42
Chapter 1
14. Find the limit: lim
x→2
1
.
x 22
(a)
(b) (d)
1
4
(e) None of these
15. Find the limit: lim 2 x→0
(c) 0
5
.
x2
(a) 7
(b) 2
(d) 0
(e) None of these
16. At which values of x is f x (c)
x2 2x 3
discontinuous?
x2
(a) 2
(b) 1, 2, 3
3
(d) 1, , 2, 3
2
(e) None of these
17. Let f x (c) 1
1
and gx x2 5. Find all values of x for which f g(x) is discontinuous.
x1
(a) 1
(b) 1, ± 5
(d) 2, 2
(e) None of these
(c) ± 5
18. Determine the value of c so that f x is continuous on the entire real line when f x (a) 0
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(d)
Test Bank
5
6
(b)
6
5
(c) 1
(e) None of these
19. Find all vertical asymptote(s) of f x x3
.
x2
(a) x 2, x 3
(b) x 2
(d) x 1
(e) None of these
20. Find all vertical asymptote(s) of gx (c) x 3
2x 3
.
x3
2x2
3
2
3
(a) x , x 1
2
(b) x (d) y 1
(e) None of these
(c) x 1
xcx2,3,
x ≤ 5
.
x > 5
43
44
Chapter 1
Limits and Their Properties
Test Form B
Name
__________________________________________
Date
Chapter 1
Class
__________________________________________
Section _______________________
____________________________
1. Find the limit: lim 2x2 1.
x→3
(c) 17
(a) 37
(b) 19
(d) ± 2
(e) None of these
2. Given lim 2x 1 3. Find such that 2x 1 3 < 0.01 whenever 0 < x 1 < .
x→1
(a) 3
(b) 0.05
(d) 1
(e) None of these
3. Use the graph at the right to find lim f x for f x x→1
(a) 0
(c)
(c) 0.005
1
.
x1
y
(b) 1
(d) The limit does not exist.
1
x
−1
(e) None of these
1
2
3
4
x2 2x 3
.
x→1
x2 1
4. Find the limit: lim
(b) 1
(d) The limit does not exist.
(e) None of these
(c)
5. Find the limit: lim 9 x2.
x→3
(a) 0
(b) 6
(d) The limit does not exist.
(e) None of these
(c) 32
6. If lim f x 12 and lim gx 23, find lim f xgx .
x→c
(a)
x→c
x→c
(b) 13
1
6
(d) The limit does not exist.
(c) 1
(e) None of these
x2 5x 6
.
x→1
x1
7. Find the limit: lim
(a) 0
(d)
(b) 7
(e) None of these
(c) © Houghton Mifflin Company. All rights reserved.
(a) 0
Chapter 1
8. Find the limit: lim
x→2
x2
.
x2
(a) 0
(b) 1
(d) The limit does not exist.
(e) None of these
9. Find the limit: lim
x 9 3
x
x→0
.
(a) 0
(d)
(b) 1
1
3
(c)
(e) None of these
10. Find the limit: lim csc
x→5
x
.
4
(b) 1
(a) 1
(d) (c) 2
1
(c) 2
(e) None of these
2
1 cos2 x
.
x→0
x
11. Find the limit: lim
(a) 1
(b) 0
(d) The limit does not exist.
(e) None of these
(c)
12. Find the limit: lim 2x 3.
x→2
(a) 1, 1
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(d)
(b) 1
1
2
(c) 1
(e) None of these
1
13. Find the limit: lim .
x→0 x
(a) (b) 0
(d) The limit does not exist.
(e) None of these
14. Find the limit: lim
x→1
5
.
x 12
(a) 0
(b) (d) (e) None of these
15. Find the limit: lim 2 x→1
(c) (c)
5
4
5
.
x 12
(a) (b) (d) 2
(e) None of these
(c) 3
Test Bank
45
Chapter 1
Limits and Their Properties
16. At which values of x is f x x4
discontinuous?
x2 x 2
(a) 4
(b) 1, 2, 4
(d) 1, 2, 4, 2
(e) None of these
17. Let f x (c) 1, 2
1
and gx x 1. Find all values of x for which f g(x) is discontinuous.
x
(a) 0
(b) 1
(d) 1, 1
(e) None of these
(c) 0, 1
18. Determine the value of c so that f x is continuous on the entire real line when f x (a) 4
(b) 4
(d) 1
(e) None of these
19. Find all vertical asymptote(s) of f x (a) x 2
(d) x 1
2
x2x3,c,
x ≤ 1
.
x > 1
(c) 0
2x 1
.
x3
1
(b) x , x 3
2
(c) x 3
(e) None of these
20. Find all vertical asymptote(s) of f x x2
.
x2 4
(a) x 2, x 2
(b) x 2
(d) x 2
(e) None of these
(c) x 0
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46
Chapter 1
Test Bank
Test Form C
Name
__________________________________________
Date
Chapter 1
Class
__________________________________________
Section _______________________
____________________________
A graphing calculator is needed for some problems.
1. Use a graphing calculator to graph the function: f x x2 4x and then estimate lim f x (if it exists).
x→2
(a) 0
(b) 12
(c) 4
(d) 12
(e) None of these
2. Use the graph to find lim f x (if it exists).
y
x→1
(a) 1
2
(b) 2
1
y = f ( x)
(c) The limit does not exist.
x
−2
1
−2
(e) 3
3. Find the limit: lim
x→1
x2 x 2
.
x3
(a) 3
(b) 0
(d) The limit does not exist.
(e) None of these
4. Find the limit: lim
x→ 2
(b)
22
2
2
3
2
(c) 3
(e) None of these
6. Find the limit: lim
x→9
(d) 0
(c) x2
and then estimate lim f x.
x→0
x3
(b) 2
(d) 0
(a) 12
1
2
(e) None of these
5. Use a graphing calculator to graph the function f x (a) (c) sin x
.
x
(a) 0
(d)
2
−1
(d) 1
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47
x2 6x 27
.
x9
(b) The limit does not exist.
(e) None of these
(c) 3
48
Chapter 1
Limits and Their Properties
1
1
x3 3
7. Find the limit: lim
.
x→0
x
(a) 1
9
(b) 0
(d) The limit does not exist.
(c)
1
9
(c)
(e) None of these
1 cos2 x
.
x→0
x
8. Find the limit: lim
(a) 1
(b) 0
(d) The limit does not exist.
(e) None of these
9. Match the graph with the correct function.
x3
(a) f x x1
(c) f x x2
x1
2x 3
y
6
(b) f x x 3
5
(1, 4)
4
(d) f x x2 2x 3
x1
2
1
(e) None of these
−2
1
,
x3
10. Find the x-values (if any) for which f is not continuous. f x 1
,
2
−1
x
1
2
3
x ≤ 5
x > 5
1
2
(a) 5
(b)
(c) 3
(d) 3, 5
11. Determine all values of c so that the function f is continuous on , . f x (a) 3
(b) 1
(d) 1, 3
(e) 1
2x 3,
x2,
x < c
x ≥ c
(c) 3, 1
12. Use a graphing calculator to graph f x x3 2x 5. Then use this graph to find the interval for
which the Intermediate Value Theorem guarantees the existence of at least one number c in that
interval for which f c 0.
(a) 1, 1
(b) 1, 2
(d) 3, 4
(e) None of these
(c) 2, 3
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(e) None of these
Chapter 1
Test Bank
13. Use the graph to find the interval(s) for which the function f is continuous.
(a) , 2 and 2, (b) , (c) , 1 and 1, (d) 1, 2
y
6
3
(e) None of these
x
−3
3
−3
14. Find the limit:
lim
x→ 12
2x 1.
(a) 0
(b) 2
(d) The limit does not exist.
(e) None of these
15. Find the limit: lim
x→6
(c) 1
3x 18.
6x
(a) 1
(b) 1
(d) 3
(e) None of these
(c) 3
16. Which of the following statements is not true of f x x2 25?
(a) f is continuous at x 10.
(b) f is continuous on the interval , 5.
(c) f is continuous on the interval 5, .
(d) f is continuous on the interval 5, 5.
(e) f is not continuous at x 0.
17. Find the limit: lim
x→1
(a)
2
.
x1
(b) © Houghton Mifflin Company. All rights reserved.
(d) The limit does not exist.
18. Find the limit: lim
x→3
(a)
1
3
(d) 1
(c) 0
(e) None of these
x2 3x 2
.
x2 5x 6
(b) (c) (e) None of these
19. Find the vertical asymptote(s): f x x2
.
x2 3x 10
(a) x 2, x 5
(b) y 1
(d) x 5
(e) x 5
49
(c) y 0
20. f x decreases without bound as x approaches what value from the right? f x (a) 5
(b) 3
(d) 3
(e) None of these
4
x 35 x
(c) 5
6
50
Chapter 1
Limits and Their Properties
Test Form D
Name
__________________________________________
Date
Chapter 1
Class
__________________________________________
Section _______________________
____________________________
1. Calculate lim 2x2 6x 1.
x→2
2. If lim 3x 2 7, find such that 3x 2 7 < 0.003 whenever 0 < x 3 < .
x→3
3. Find the limit: lim f x if f x x→1
x2, 4,
2
x1
.
x1
x2 3x 2
.
x→1
x2 1
4. Find the limit: lim
5. Find the limit: lim 4x2 9.
x→2
6. If lim f x 2 and lim gx 3, find lim f x gx.
1
2
x→c
x→c
x→c
x2
.
x→2 x3 8
7. Find the limit: lim
x x x
8. Find the limit: lim
x
x→0
x→1
x1
.
x1
10. Find the limit: lim cot
x→5
x
.
6
x
.
x→0 sin 3x
11. Find the limit: lim
12. Find the limit: lim 2x 1.
x→2
13. Find the limit:
lim x→1
14. Find the limit: lim
x→3
1
.
x1
3
.
x2 6x 9
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9. Find the limit: lim
.
Chapter 1
15. Find the limit: lim 3x 2 x→0
1
.
x2
16. Find the value(s) of x for which f x removable or nonremovable.
17. Let f x x2
is discontinuous and label these discontinuities as
x2 4
5
and gx x4.
x1
a. Find f g(x).
b. Find all values of x for which f g(x) is discontinuous.
© Houghton Mifflin Company. All rights reserved.
x2,
18. Determine the value of c so that f x is continuous on the entire real line if f x c
,
x
19. Find all vertical asymptote(s) of f x if f x x2 3x 1
.
x7
20. Find all vertical asymptote(s) of f x if f x 2x 2
.
x 1x2 x 1
x ≤ 3
.
x > 3
Test Bank
51
52
Chapter 1
Limits and Their Properties
Test Form E
Name
__________________________________________
Date
Chapter 1
Class
__________________________________________
Section _______________________
____________________________
A graphing calculator is needed for some problems.
1. Use the graph to find lim f x (if it exists).
y
x→0
4
3
y = f ( x)
1
−2
−1
x
1
2
2. Determine whether the statement is true or false. If it is false, give an example to show that it is false.
If lim f x 9, then f 3 9.
x→3
3. Calculate the limit: lim 2x2 6x 1.
x→2
4. Find the limit: lim
x2 x 2
.
x3
5. Find the limit: lim
x
.
cos x
x→1
x→ 6. Use a graphing calculator to graph the function f x x3 x 5 and then estimate lim f x.
x→1
x2 4
.
x2
a. Use a graphing calculator to graph the function.
b. Use the graph to estimate lim f x.
x→2
c. Find the limit by analytical methods.
8. Let f x x3 2x2
.
x2
a. Find lim f x (if it exists).
x→2
b. Identify another function that agrees with f x at all but one point.
c. Sketch the graph of f x.
9. Find the limit: lim
x→0
10. Find the limit: lim
x→0
x x 2 x 2
x
.
x x2 2x x x2 2x
.
x
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7. Let f x Chapter 1
Test Bank
11. Find the x-values (if any) for which f is not continuous.
f x 3x2x 2,3x 6,
2
x < 1
x ≥ 1
12. Determine all values of c so that the function f is continuous on , .
xx , 1,
2
f x x < c
x ≥ c
13. Find the interval(s) for which the function g shown in the graph is continuous.
g(x) = tan
y
x
2
9
6
3
x
2π
−3
−6
−9
14. Use the Intermediate Value Theorem to show that the function f x x 4 2x2 3x has a zero
in the interval 2, 1.
15. Use a graphing calculator to graph f x the function is not continuous.
16. Let f x x2 1,
2x 3,
x ≤ 0
. Find each limit (if it exists).
x > 0
a. lim f x
b. lim f x
x→0
© Houghton Mifflin Company. All rights reserved.
x2
. Then use the graph to determine x-values at which
x2 4
x→0
17. Use a graphing calculator to find the limit: lim
x→0
18. Find all vertical asymptotes of f x 19. Find the limit: lim
x→3
20. Find the limit: lim
x→1
1
.
x3
2
.
1 x2
c. lim f x
x→0
sin 3x
. Then verify your answer analytically.
x
x2 x 2
.
x2 x 6
53