Deck 2: Limits and Their Properties

Question 1

Suppose that . Find the following limit. ​ ​

A) -1
B) 1
C) 3
D) 0
E) 2

Answer 1

B

Question 2

Find the limit (if it exists). ​ ​

A) 0
B)
C)
D)
E) Limit does not exist.

Answer 2

D

Question 3

Suppose that . Find the following limit. ​ ​

A) -5
B) -55
C) 55
D) -11c
E) -11

Answer 3

C

Question 4

Find the following limit (if it exists). Write a simpler function that agrees with the given function at all but one point. ​ 7
​

A)
B)
C)
D)
E) does not exist

Question 5

Use a graphing utility to graph the function and determine the following one-sided limit. ​ ​

A)
B) 6
C) -6
D)
E) 0

Question 6

A ring has a inner circumference of 9 centimeters. If the ring's inner circumference can vary between 8.5 centimeters and 9.5 centimeters how can the radius vary? Round your answer to five decimal places. ​

A) Radius can vary between 7.32046 centimeters and 9.14424 centimeters.
B) Radius can vary between 1.64488 centimeters and 1.73895 centimeters.
C) Radius can vary between 1.35282 centimeters and 1.51197 centimeters.
D) Radius can vary between 2.70563 centimeters and 3.02394 centimeters.
E) Radius can vary between 0.93239 centimeter and 1.93239 centimeters.

Question 7

Find the constant a and b such that the function ​ ​
Is continuous on the entire real line.
​

A) a = 3, b = 0
B) a = 3, b = -3
C) a = -3, b = -3
D) a = -3, b = 3
E) a = 3, b = 3

Question 8

Find the limit. ​ ​

A) 1
B)
C) 0
D)
E) -1

Question 9

Find where . ​

A) 1
B) 4
C) -5
D) 1
E) Limit does not exist.

Question 10

A 25-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate of where x is the distance between the base of the ladder and the house. Find the limit of r as . ​ ​

A)
B) 50
C) 0
D)
E) 25

Question 11

Find the vertical asymptotes (if any) of the function . ​

A)
B)
C)
D)
E) no vertical asymptotes

Question 12

A sphere has a volume of 4.12 cubic inches. What is the radius of the sphere? Round your answer to four decimal places. ​

A) 0.9945 inch
B) 1.5787 inches
C) 0.9918 inch
D) 1.9835 inches
E) 1.7988 inches

Question 13

A long distance phone service charges $0.35 for the first 8 minutes and $0.05 for each additional minute or fraction thereof. Use the greatest integer function to write the cost C of a call in terms of time t (in minutes). ​

A)
B)
C)
D)
E)

Question 14

Suppose that and . Find the following limit. ​ ​

A)
B)
C)
D)
E) does not exist

Question 15

Find all the vertical asymptotes (if any) of the graph of the function . ​

A) x = -1
B) x = 5
C) x = 1, -1
D) x = 1
E) no vertical asymptotes

Question 16

Use the graph to find the limit. ​ ​

A)
B)
C)
D)
E) The limit does not exist.

Question 17

Let ​ Determine the following limit.

Question 18

Determine the limit (if it exists). ​ ​

A) 0
B) 1
C) 2
D) ∞
E) does not exist

Question 19

Find the x-values (if any) at which the function is not continuous. Which of the discontinuities are removable? ​

A) 10 and -10, removable
B) discontinuous everywhere 0 00
C) continuous everywhere 0 00
D) 10 and -10, not removable
E) 0, removable

Question 20

Let ​ Determine the following limit.

Question 21

Find the limit (if it exists). Note that represents the greatest integer function. ​ ​

A) -109
B) 119
C) -119
D) 109
E) does not exist

Question 22

Complete the table and use the result to estimate the limit. ​ ​ ​

A)
B)
C)
D)
E)

Question 23

Find the value of c guaranteed by the Intermediate Value Theorem. ​ ​

A) 7
B) 2
C) 1
D) 5
E) 6

Question 24

Suppose that and . Find the following limit. ​ ​

A) 0
B) -4
C) -9
D) -17
E) 52

Question 25

Find the following limit if it exists: . Use when appropriate. ​

A)
B) -4
C) 1
D)
E) does not exist

Question 26

A petrol car is parked 35 feet from a long warehouse (see figure). The revolving light on top of the car turns at a rate of revolution per second. The rate at which the light beam moves along the wall is ft/sec. Find the rate r when is . ​ ​

A) ft/sec
B) ft/sec
C) ft/sec
D) ft/sec
E) ft/sec

Question 27

Determine the limit (if it exists). ​ ​

A) 8
B) 1
C) 0
D) -2
E) does not exist

Question 28

A petrol car is parked 65 feet from a long warehouse (see figure). The revolving light on top of the car turns at a rate of revolution per second. The rate at which the light beam moves along the wall is ft/sec. Find the limit of r as . ​ ​

A)
B)
C) 0
D) 65
E)

Question 29

Find the lmit. ​ ​

A)
B)
C)
D)
E) does not exist

Question 30

Find the x-values (if any) at which is not continuous. ​

A) f(x) is not continuous at x = 0 and f(x) has a removable discontinuity at x = 0.
B) f(x) is not continuous at x = 0, -2 and both the discontinuities are nonremovable.
C) f(x) is not continuous at x = -2 and f(x) has a removable discontinuity at x = -2.
D) f(x) is not continuous at x = 0, -2 and f(x) has a removable discontinuity at x = 0.
E) f(x) is continuous for all real x.

Question 31

Find the limit. ​ ​

A)
B)
C)
D)
E)

Question 32

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 wherexand are measured in miles. Find the rate of change of elevation when x= 3.

A) precalculus, 0.06
B) calculus, 0.18
C) calculus, 0.36
D) calculus, 0.06
E) precalculus, 0.18

Question 33

A 30-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate of , where x is the distance between the base of the ladder and the house. Find the rate r when x is 18 feet. ​ ​

A) r = ft/sec
B) r = ft/sec
C) ft/sec
D) r = ft/sec
E) r = ft/sec

Question 34

Use the rectangles in the following graph to approximate the area of the region bounded by . ​

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

Question 35

Let and . Find the limit. ​ ​

A)
B)
C)
D)
E)

Question 36

Find the limit (if it exists). ​ ​

A) Limit does not exist.
B) 0
C) 3
D) 4
E) 9

Question 37

Find the limit. ​ ​

A) 1
B) 0
C) -1
D)
E)

Question 38

Determine the following limit.

Question 39

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), (7,8), (15,0).
​ ​

A) precalculus , 120
B) calculus , 180
C) precalculus , 60
D) precalculus , 180
E) calculus , 120

Question 40

Find the x-values (if any) at which the function is not continuous. Which of the discontinuities are removable? ​

A) no points of discontinuity
B) x = -9 (not removable), x = -5 (removable)
C) x = -9 (removable), x = -5 (not removable)
D) no points of continuity
E) x = -9 (not removable), x = -5 (not removable)

Question 41

Suppose that and . Find the following limit. ​ ​

A) -3
B) 5
C) 24
D) -11
E) 0

Question 42

Determine the following limit.

Question 43

Use the rectangles in the graph given below to approximate the area of the region bounded by . Round your answer to three decimal places. ​

A) 2.481 units²
B) 6.371units²
C) 3.585units²
D) 6.872units²
E) 6.903units²

Question 44

Use the graph as shown to determine the following limits, and discuss the continuity of the function at x= 4. ​
(i) (ii) (iii) ​ ​

A) 1, 1, 1, not continuous
B) 2, 2, 2, continuous
C) 1, 1, 1, continuous
D) 2, 2, 2, not continuous
E) 0, 0, 0, not continuous

Question 45

Find the limit. ​ ​

A)
B)
C) -12
D) 6
E) 12

Question 46

Find the limit (if it exists). ​ ​

A)
B)
C)
D)
E) does not exist

Question 47

Find the limit (if it exists). ​ ​

A)
B)
C)
D)
E)

Question 48

Suppose that and . Find the following limit. ​ ​

A) 9
B) -3
C) -21
D) -12
E) 108

Question 49

Use the graph as shown to determine the following limits, and discuss the continuity of the function at x= -4. ​
(i) (ii) (iii) ​

A) 2, 2, 2, continuous
B) 1, 1, 1, not continuous
C) 2, 2, 2, not continuous
D) -4, -4, -4, continuous
E) 1, 1, 1, continuous

Question 50

Find the x-values (if any) at which is not continuous. ​

A) f(x) is not continuous at x = -2 and the discontinuity is nonremovable.
B) f(x) is not continuous at x = 0 and the discontinuity is removable.
C) f(x) is continuous for all real x.
D) f(x) is not continuous at x = -2 and the discontinuity is removable.
E) f(x) is not continuous at x = 0, 2 and x = 0 is a removable discontinuity.

Question 51

Find the limit (if it exists). ​ ​

A)
B)
C) 0
D)
E) Limit does not exist

Question 52

Find the limit. ​ ​

A)
B) ​
C)
D)
E)

Question 53

Complete the table and use the result to estimate the limit. ​ ​ ​

A)
B)
C)
D)
E)

Question 54

Find the value of c guaranteed by the Intermediate Value Theorem. ​ ​

A) 1
B) 4
C) 6
D) 2
E) 5

Question 55

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 where x and are measured in miles. Find the rate of change of elevation when x = 3.
​

A) calculus, 0.72
B) precalculus, 0.12
C) calculus, 0.12
D) precalculus, 0.72
E) precalculus, 0.37

Question 56

Determine the following limit.

Question 57

Use a graphing utility to graph the function and determine the one-sided limit . ​

A)
B)
C) 0
D) 3
E) 2

Question 58

Find the x-values (if any) at which the function f(x) = 11x² - 5x - 5 is not continuous. Which of the discontinuities are removable? ​

A) x = 2, removable
B) x = 0, removable
C)
D) continuous everywhere
E)

Question 59

Find the vertical asymptotes (if any) of the function . ​

A) x = 2
B) x = 6
C) x = -6
D) x = 12
E) x = -2

Question 60

Find the limit. ​ ​

A)
B)
C)
D)
E)

Question 61

Find the limit. ​ ​

A)
B)
C)
D)
E)

Question 62

A sphere has a volume of 4.0 cubic inches. If the sphere's volume can vary between 3.2 cubic inches and 5.5 cubic inches , how can the radius vary? Round your answer to five decimal places. ​

A) Radius can vary between 0.91416 inch and 1.09503 inches.
B) Radius can vary between 1.45113 inches and 1.73825 inches.
C) Radius can vary between 0.18475 inch and 2.48475 inches.
D) Radius can vary between 1.58533 inches and 2.07839 inches.
E) Radius can vary between 0.87404 inch and 1.14587 inches.

Question 63

Find the limit (if it exists). Note that represents the greatest integer function. ​ ​

A) 13
B) Limit does not exist.
C) 12
D) 0
E) 11

Question 64

Let and . Find the limit. ​ ​

A)
B)
C)
D)
E)

Question 65

Find the constant a such that the function ​ ​
Is continuous on the entire real line.
​

A) 1
B) -7
C) 7
D) 14
E) -14

Question 66

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 13 seconds by an object moving with a velocity of feet per second.
​

A) calculus, 162.3521 ft
B) precalculus, 163.7021 ft
C) calculus, 158.9412 ft
D) precalculus, 158.9412 ft
E) precalculus, 162.3521 ft

Question 67

Determine the limit (if it exists). ​ ​

A) 0
B) 56
C) 11
D) 28
E) does not exist

Question 68

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) calculus , 15
B) precalculus , 15
C) precalculus , 18
D) calculus , 22
E) precalculus , 22
​

Question 69

Use the graph to determine the following limits, and discuss the continuity of the function at x = -4. (i) (ii) (iii) ​

A) 2, -2, does not exist, not continuous
B) 2, 0, does not exist, not continuous
C) 0, 2, does not exist, not continuous
D) -4, 0, does not exist, not continuous
E) 0, 2, 0, continuous

Question 70

Consider the length of the graph of from to . Approximate the length of the curve by finding the sum of the lengths of fiveline segments, as shown in following figure. Round your answer to two decimal places. ​

A) 7.76
B) 9.77
C) 7.07
D) 9.9
E) 8.76

Question 71

Find all values of c such that f is continuous on (-∞,∞). ​ ​

A) c = 2
B) c = 0
C)
D)
E)

Question 72

Find all vertical asymptotes (if any) of the function ​

A) x = 1, 5
B) x = 1, 5, -4
C) x = -1, -5
D) x = 5
E) x = -5

Question 73

Complete the table and use the result to estimate the limit. ​ ​ ​

A)
B)
C)
D)
E)

Question 74

Use the rectangles in the following graph to approximate the area of the region bounded by . ​ ​

A) 3.9485
B) 2.6323
C) 1.9742
D) 1.4807
E) 0.9871

Question 75

Find the limit (if it exists). ​ ​

A) 10
B)
C)
D)
E) .

Question 76

Find the limit (if it exists). ​ ​

A)
B) 0
C) Limit does not exist.
D)
E)

Question 77

Let and . Find the limit. ​ ​

A)
B)
C)
D)
E)

Question 78

Find the limit. ​ ​

A)
B)
C)
D)
E)

Question 79

Find all the vertical asymptotes (if any) of the graph of the function . ​

A) x = -2
B) x = 4
C) x = 0
D) x = 4, x = 0
E) no vertical asymptotes

Question 80

A ring has a inner circumference of 2 centimeters. What is the radius of the ring? Round your answer to four decimal places. ​

A) 0.1592 centimeter
B) 0.6366 centimeter
C) 0.3183 centimeter
D) 0.7979 centimeter
E) 0.4053 centimeter