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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

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 2 f ( x) = 0.09 (18 x − x ) 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

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

Chapter 1: Limits and Their Properties

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 = π .

A) 1.4221 B) 3.7922 C) 0.9481 D) 1.8961 E) 1.2704

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 = −

π

2

, and x =

π

2

.

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. x–5 lim 2 x →5 x – 20 x + 75

x 4.9 4.99 4.999 5.001 5.01 5.1 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 x –7.1 –7.01 –7.001 –6.999 –6.99 –6.9 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 x–5 x 4.9 4.99 4.999 5.001 5.01 5.1 f(x) A) 0.114375 B) 0.094375 C) –0.145625 D) –0.015625 E) –0.125625

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 x –0.1 –0.01 –0.001 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 x –0.1 –0.01 –0.001 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

A) 7 B) 5 C) 2 D) 3 E) Does not exist

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 ( x2 + 3 ) x →2

A) Does not exist B) 2 C) 3 D) 7 E) 0

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) = –3 x – 5 and g ( x) = x2 . Find the limits: (a) lim f ( x) x→ –4

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

(b) lim g( x) x→ –2

(c) lim g ( f (x )) x →4

D) 12 , –2 , 25 E) 7 , 4 , 289

Chapter 1: Limits and Their Properties

22. Let f ( x ) = x 2 + 1 and g ( x) = 3x . Find the limits: (b) li m g (x )

(a) lim f ( 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 + x2 and g ( x ) = x + 3 . Find the limits:

(b) li m g (x )

(a) lim f ( x)

(c) lim g ( f ( x ))

x →5

x→ 2

A) 9, 8, 33

x→ 5

B) 2, 5, 5

C) 6,5, 30

D) 4, 3, 8

E) 6, 5, 33

24. Let f ( x ) = 5x 2 + 3x – 2 and g ( x ) = 3 x – 4 . Find the limits: (b) li m g (x )

(a) lim f ( x) x→ 2

(c) lim g ( f ( x ))

x→ 2

x→ 5

A) 24, 3 2, – 3 134 B) 28, – 3 2, – 3 134

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

C) 28, 3 2, 3 134 25. Find the limit: lim3π sin x

x→

4

A) −

21/ 2 2

B)

21/ 2 2

C)

2 –1/ 2 4

D) Does not exist E) −

2 –1/ 2 4

26. Find the limit:  πx  lim cos   x →5  6  1/ 2 –1/ 2 3 3 A) B) 4 2

1/ 2

C) –

3

2

–1/ 2

D) 0 E) –

3

4

27. Find the limit:  5x  lim tan   x →π  6  –1/ 2 –1/ 2 B) 3 A) –3

1/ 2 C) 6

D) Does not exist E) –61/ 2

Chapter 1: Limits and Their Properties

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

x →c

lim  f (x ) x →c

A) –11

 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

x→ c

lim [ f (x ) + g (x )] x →c

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

x

c

lim [ f (x ) − g (x )] x →c

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

x →c

lim [ –9 g(x ) ] x →c

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

x →c

lim [ f (x )g (x ) ] 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 ) 2 3 B) 54 C) A) 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 – 8x + 64 B) 64 , x 2 + 8x + 64 C) 64 , x 2 – 8x – 64 lim

D) Limit does not exist. E) –192 , x 2 – 8x + 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 x2 – 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

36. Find the limit (if it exists): 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) 2 – ( x + ∆x ) – 6 − ( x 2 – x – 6 )

∆x →0

∆x

1 1 A) x 3 – x 2 – 6 x 3 2

B) x3 – x2 – 6x

C) 0 D) 2 x – 1

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): 5

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

E) x 2 – x – 6

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

(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) = –3x 2 – 7 x – 11 is not continuous. Which of the discontinuitites are removable? A) Continuous everywhere D) 7 x = – . Not removable. 6 B) x = –11 . Removable E) both B and C 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.

Chapter 1: Limits and Their Properties

47.

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

x –5 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 

is continuous on the entire real line. 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

–9.1

–9.01

–9.001

x –8.999 f(x) A) lim− f ( x) = ∞

–8.99

–8.9

–8.5

x →–9

B) C)

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.

x2 − 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) =

2

x + 8 x + 12 . 3 x + 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) . D) No vertical asymptotes A) 2k + 1 π ( k = 0, ± 1, ± 2, ...) x= 30 E) B) 2k 2k + 1 π (k = 0, ± 1, ± 2, ...) x = π ( k = 0, ± 1, ± 2, ...) x= 30 15 C) 2k π (k = 0, ± 1, ± 2, ...) x= 15 54. Find the limit: x–8 – x+ 6 A) – ∞ B) ∞ lim+ x →6

C) 0 D) –1 E) 1

55. Find the limit: lim

x →12

x 2 – 12x ( x 2 +144 ) (x – 12 )

A) –24 B) –

1 24

C) 24 D)

1 24

56. Find the limit: 1  lim−  x 4 –  x →0  x A) 0 B) –∞

C) 1 D) –1 E) ∞

E) 12

Chapter 1: Limits and Their Properties

Answer Key 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:

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

Chapter 1: Limits and Their Properties

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

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:

1.4 1.4 1.4 1.4 1.5 1.5 1.5 1.5 1.5 1.5 1.5...