Deck 2: Limits and Continuity

Question 1

An object moves along the x-axis so that at time t seconds (where t $\ge$ 0) it is x = m to the right of the origin. By calculating the average velocity of the object over time intervals [4, 4 + h] for h = 0.1, h = 0.01, h = 0.001, and h = 0.0001, determine the velocity of the particle at time t = 4s.

A) 0.2499 m/s
B) 0.2498 m/s
C) 0.2485 m/s
D) 0.2000 m/s
E) 0.2501 m/s

Answer 1

0.2499 m/s

Question 2

An object moves along the x-axis so that its velocity at time t seconds is v(t) = m/s. What is the average acceleration (i.e., average rate of change of the velocity) of the object over the time interval[2, 2 + h]? What is the acceleration of the object at time t = 2s?

A) - m/ , - m/
B) m/ , m/
C) m/ , m/
D) - m/ , - m/
E) m/ , m/

Answer 2

- m/ , - m/

Question 3

The temperature of an object at time t minutes is degrees. What is the rate of change of the temperature over the time interval [t, t + h]? How fast is the temperature changing at t = 1?

A) 10 degrees/minute, increasing at 5 degrees/minute
B) -10 degrees/minute, decreasing at 5 degrees/minute
C) degrees/minute, increasing at 5 degrees/minute
D) -10 degrees/minute, decreasing at 5 degrees/minute
E) 10 degrees/minute, decreasing at 5 degrees/minute

Answer 3

-10 degrees/minute, decreasing at 5 degrees/minute

Question 4

In the following figure the curve is the graph of a quantity y that is a function of time t. Also shown is its tangent line at t = 100. What is the average rate of change of y from t = 20 to t = 160? What is the approximate rate of change of y with respect to t at t = 100?

A) ,
B) ,
C) ,
D) ,
E) none of the above

Question 5

Evaluate the following limits:
(i)
(ii)

A) (i) (ii) -6
B) (i) (ii) 6
C) (i) 1 (ii) 1
D) (i) (ii) does not exist
E) (i) (ii) does not exist

Question 6

Find the limit of f(x) as x approaches 2 if f is defined byf(x) =

A) 1
B) 2
C) 3
D) 0
E) does not exist

Question 7

Evaluate .

A)
B) 0
C)
D) -
E) - $\infty$

Question 8

Evaluate .

A)
B) 2
C)
D) 4
E) does not exist

Question 9

Evaluate .

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

Question 10

Evaluate .

A) 4/7
B) 2/5
C) 0
D) 1/2
E) does not exist

Question 11

Evaluate .

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

Question 12

Evaluate .

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

Question 13

If = - 5, find f(x).

A) 14
B) - 14
C) 0
D) $\infty$
E) - 19

Question 14

Evaluate .

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

Question 15

Let f(x) = and g(x) = - 2. Which of the following is true?

A) f = 0
B) g = -2
C) f = 0
D) g = -2
E) g = -2

Question 16

Evaluate .

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

Question 17

Evaluate .

A) 3
B) 1
C) 4
D) 0
E) does not exist

Question 18

Evaluate .

A)
B)
C)
D) $\infty$
E) does not exist

Question 19

Find the following limit: .

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

Question 20

Evaluate .

A) - 5
B) 2
C) 5
D) - $\infty$
E) - 3

Question 21

Evaluate .

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

Question 22

Evaluate .

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

Question 23

Let f(x) = and g(x) = 14 - . Compute .

A) -
B)
C) -
D)
E)

Question 24

If = -32, find k.

Question 25

Evaluate .

A) - $\infty$
B) -
C)
D)
E) 0

Question 26

Evaluate .

A) 3
B) -3
C)
D) 1
E) does not exist

Question 27

Evaluate the following limit: x + .

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

Question 28

Evaluate the following limit: x .

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

Question 29

Compute the following limit:.

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

Question 30

Question 31

Question 32

Question 33

If f(x) = L, then = .

Question 34

Evaluate .

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

Question 35

Evaluate ( - x).

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

Question 36

Evaluate .

A) + $\infty$
B) 14
C) - $\infty$

D) -14
E) 0

Question 37

Let g(x) = .Evaluate g(x).

A) $\infty$
B) -3
C) - $\infty$
D) 1
E) none of the above

Question 38

Evaluate .

A) 0
B) 1
C) $\infty$
D) -1
E) 3

Question 39

Evaluate (3 - + x - 7).

A) - $\infty$
B) 0
C) 1
D) $\infty$
E) none of the above

Question 40

Question 41

Evaluate .

A) $\infty$
B) - $\infty$
C) 1
D) 0
E) does not exist

Question 42

Find all values of the real number k so that = .

A) 0
B) -
C) ±
D) ± 3
E) -3

Question 43

Evaluate .

A)
B) 0
C) + $\infty$
D) -
E) - $\infty$

Question 44

Find .

A) 0
B) $\infty$
C)
D) 2
E) does not exist

Question 45

Evaluate .

A) 0
B) 1
C) -1
D) $\infty$
E) - $\infty$

Question 46

Let f(x) = . Find f(x).

A) + $\infty$
B) 0
C) 1
D) - $\infty$
E) -1

Question 47

Find the limit.

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

Question 48

Evaluate .

A) 1
B) - $\infty$
C) $\infty$
D) 0
E) none of the above

Question 49

Evaluate .

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

Question 50

Evaluate .

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

Question 51

Evaluate .

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

Question 52

Evaluate .

A) $\infty$
B) - $\infty$
C) 1
D) -1
E) 2

Question 53

Evaluate .

A) $\infty$
B) 0
C) 1
D) -1
E)

Question 54

Evaluate . sin x

A) $\infty$
B) - $\infty$
C) 0
D) 1
E) does not exist

Question 55

Evaluate .

A) $\infty$
B) 1
C) - $\infty$
D) -1
E) 2

Question 56

Evaluate .

A) $\infty$
B) 1
C) 0
D) - $\infty$
E) 2

Question 57

Evaluate .

A) $\infty$
B) - $\infty$
C) 0
D) -1
E) -2

Question 58

Evaluate .

A) $\infty$
B) - $\infty$
C) 2
D) -
E) none of the above

Question 59

Evaluate .

A) -
B) + $\infty$
C) -
D) 0
E) -

Question 60

Evaluate .

A) $\infty$
B) -2
C) 0
D) - $\infty$
E) $\pi$

Question 61

Evaluate .

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

Question 62

Evaluate .

A) -3
B) 3
C) 1
D) -1
E) does not exist

Question 63

Evaluate (2 tan x + 3 cot x).

A) 3
B) 2
C) $\infty$
D) - $\infty$
E) 5

Question 64

Given that = 1, evaluate (5 + 7x - 3) sin .

A) 5
B) 7
C) 0
D) -3
E) does not exist

Question 65

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

Question 66

If f(x) = $\infty$ , then f(x) does not exist.

Question 67

If f(x) = 0, then = = $\infty$

Question 68

If f(x) = 0, then = = $\infty$

Question 69

Courier rates for delivery of mail up to 200 g are given in the following chart:
Draw the graph of the cost (in dollars) of mailing a letter as a function of its mass (in grams). Where are the discontinuities of this function? Is the cost function right or left continuous at these points?

A) discontinuous at 30, 50, 100; left but not right continuous there
B) discontinuous at 30, 50, 100; right but not left continuous there
C) discontinuous at 30, 50, 100; neither left nor right continuous there
D) There are no discontinuities.
E) discontinuous at 30, 50, 100; right continuous at 30 and 50, left continuous at 100

Question 70

Let g(x) = where k is a constant real number. If g has a removable discontinuity at x = 2, then k is equal to:

A) -3
B) 4
C) ±4
D) 3
E) -4 or 3

Question 71

Let f(x) be defined by f(x) = Where, if anywhere, is f discontinuous?

A) at x = 0
B) at x = 3
C) nowhere
D) at x = -3
E) at x = 6

Question 72

Is the function f(x) = discontinuous at any point of its domain? Would your answer change if we also define f(3) = 6?

A) no, yes
B) yes, yes
C) no, no
D) yes, no
E) none of the above

Question 73

If h(x) = if x $\neq$ 1 and h(1) = 4, then h is continuous at every x.

Question 74

Let f(x) have values
Where is f discontinuous?

A) nowhere
B) at x = -5 only
C) at x = 5 only
D) at both x = -5 and x = 5
E) none of the above

Question 75

If f(x) exists and is finite, then either f is continuous at x = c or f has a continuous extension to x = c.

Question 76

The polynomial function P(x) = + - 7 - 2x + 10 has a zero in the closed interval [1 , 2].

Question 77

In which of the intervals (-2, -1), (-1, 0), (0, 1), and (1, 2) does the intermediate value theorem imply that f(x) = 2 - 4 + 5x - 4 must have a zero?

A) (1, 2)
B) (0, 1)
C) (-1, 0)
D) (-2, -1)
E) none of the above

Question 78

For what value of the constant c is the function f(x) = continuous everywhere?

A)
B)
C) 1
D) 0
E) none of the above

Question 79

Given f(x) = find the value of the constant real number k such that f is continuous at x = k.

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

Question 80

A) only I
B) only II
C) only I and II
D) only II and III
E) I, II, and III