Deck 2: Limits

Question 1

Determine the one-sided limits at of the function shown in the figure and state whether the limit exists at these points.

Answer 1

Left-sided
Right-sided
Limit
1
3
4
Does not exist
2
5
5
Exists (5)
3
3.5
5
Does not exist
4
3
1.5
Does not exist

Question 2

Let denote the slope of the line segment connecting the origin to the point on the graph of the equation . Calculate the average rate of change of for

Answer 2

Question 3

The electrical field due to an infinite rod at a point at distance from the rod is perpendicular to the rod and has a magnitude of ( is a constant and is the longitudinal charge density).
Find the average rate of change of the field along the interval .

Answer 3

Question 4

The volume of a sphere of radius is . What is the average rate of change of the volume when the radius increases from to ?

Question 5

Determine the one-sided limits at of the function shown in the figure and state whether the limit exists at these points.

Question 6

The graph of a function is shown in the figure.
Determine the following limits or state that the limit does not exist (if the limit is infinite, write or :

Question 7

Let denote the slope of the line segment connecting the origin to the point on the graph of the semi-ellipse . Calculate the average rate of change of for

Question 8

Let denote the slope of the line segment connecting the origin to the point on the graph of the equation . Calculate the average rate of change of for

Question 9

Determine the one-sided limits at of the function shown in the figure and state whether the limit exists at these points.

Question 10

Determine and for the function shown in the figure.

Question 11

Consider the function for .

A) Write in piecewise form. What is for positive integers ?
B) Find and .
C) For which values of does the limit fail to exist?

Question 12

The flight-time of a shell shot at an angle and initial velocity is . Compute the average rate of change of the flight time for in the interval .

Question 13

Consider the function for .

A) Write in piecewise form.
What is for positive integers ?
B) Determine and .
C) For which values of does exist?

Question 14

A balloon is blown up and takes the shape of a sphere. What is the average rate of change of the surface area of the balloon as the radius increases from 3 cm to 4 cm?

Question 15

The electrical field caused by an electrical charge at a point at distance is ( is a constant).
Find the average rate of change of the field along the interval .

Question 16

The position of a particle is given by . Compute the average velocity over the time interval . Estimate the instantaneous velocity at .

Question 17

Let denote the slope of the line segment connecting the origin to the point on the graph of the equation . Calculate the average rate of change of for

Question 18

The volume of a cone of radius and height is .
What is the average rate of change of if the radius increases from 1 to 3 and the height remains unchanged?

Question 19

The greatest integer function is defined by , where is the unique integer such that .
The graph of is shown in the figure.

A) For which values of does exist?
B) For which values of does exist?
C) For which values of does exist?

Question 20

The potential energy of a pendulum of length 1 and mass 2, relative to its rest position is .
Compute the average rate of change of the potential energy over the angle interval .

Question 21

Let , , and . To prove that if and exist then also exists, we should use:

A) The Product Rule applied to and .
B) The Quotient Rule applied to and .
C) The Sum Rule applied to and .
D) The statement is not true.
E) A and C are correct.

Question 22

Let , be functions and let . Consider the following statement:
If and exist then also exists.
To prove this statement we should use:

A) The statement is not true.
B) The Product Rule applied to and .
C) The Quotient Rule applied to and .
D) The Sum Rule applied to and .
E) None of the above.

Question 23

Evaluate the limits using the Limit Laws:

A)
B)
C)
D)

Question 24

If and , calculate the following limits if possible. If not, state not possible. Assume and are nonzero real numbers.

A)
B)
C)

Question 25

Which of the following functions are examples of the existence of the limit , although the limits of and as do not exist.

A)
B)
C)
D)
E)

Question 26

Evaluate the limits using the Limit Laws:

A)
B)
C)
D)

Question 27

Compute the following one-sided limits:

A)
B)
C)

Question 28

Determine the points at which the following functions are not continuous and state the type of discontinuity: removable, jump, infinite, or none of these.

A)
B)
C)
D)

Question 29

Find the values of which make the function continuous:

Question 30

Evaluate the limits using the Limit Laws:

A)
B)
C)

Question 31

Evaluate the limits using the Limit Laws:

A)
B)
C)

Question 32

Determine whether the function is left or right continuous at the points of discontinuity:

A)
B)

Question 33

At each point of discontinuity state whether the function is left or right continuous:

Question 34

Let . Determine whether each of the following statements is always true, never true, or sometimes true.

A)
B)
C)
D)

Question 35

If and then does not converge to a finite limit as .
For proving, we assume that exists and is finite. Then
By the Quotient Rule and by the Product Rule .
Which of the statements below completes the proof?

A) From , it follows that 1=0, which is a contradiction.
B) From , we can conclude that , which contradicts our assumption.
C) From , we can conclude that , which contradicts our assumption.
D) From , we can conclude that , which contradicts our assumption.
E) A and C are correct.

Question 36

Find the points of discontinuity and state their type

A)
B)
C)

Question 37

The following functions are examples of the existence of the limit although the limits and do not exist.

A)
B)
C)
D)
E)

Question 38

Determine whether the following statement is correct. If yes, prove it; otherwise give a counterexample
If , then exists.

Question 39

Find such that exists and compute the limit

Question 40

Let be the following function defined for : where where and where .
Write piecewise, sketch its graph, and determine the points where the limit of does not exist. Find the one-sided limits at these points.

Question 41

At each point of discontinuity state whether the function is left or right continuous.

A)
B)

Question 42

Evaluate the limit or state that it does not exist

A)
B)
C)

Question 43

Find the values of and which make the following function continuous:

Question 44

Find the value of for which the limit exists and compute the limit

Question 45

Evaluate the limit

Question 46

Evaluate the limit or state that it does not exist

A)
B)
C)

Question 47

Sketch the graph of a function that satisfies

Question 48

Let Find . at .

Question 49

Evaluate the limit or state that it does not exist

A)
B)
C)

Question 50

Determine the value of for which the limit exists and find the limit

Question 51

Find the constants and which make the function continuous:

Question 52

Evaluate the limit or state that it does not exist

A)
B)
C)

Question 53

At each point of discontinuity state whether the function is left continuous, right continuous, or neither

A)
B)

Question 54

Evaluate the limits in terms of the constants involved

A)
B)

Question 55

Find all values of and which make the following function continuous:

Question 56

Let be the following function: The function is continuous for the following function

A) if .
B) if
C) if if
D) if if
E) A and C are correct.

Question 57

Determine the points where the function is not continuous and state the type of discontinuity: removable, jump, infinite, or none of these:

A)
B)
C)
D)

Question 58

Let be a discontinuous function. Can you find a continuous function such that is continuous? Explain.

Question 59

Evaluate the limit or state that it does not exist

A)
B)
C)

Question 60

Determine the points where the function is not continuous and state the type of the discontinuity: removable, jump, infinite, or none of these.

A)
B)
C)

Question 61

Use the Squeeze Theorem to evaluate the limit

Question 62

Compute the following limits:

A)
B)
C)

Question 63

Show that for all .
Use the above inequality and the Squeeze Theorem to evaluate .

Question 64

Calculate the following limits:

A)
B)
C)

Question 65

Find .

Question 66

Calculate the limits

A)
B)
C)

Question 67

Evaluate the limits

A) (Hint: Factor the denominator)
B) (Hint: Factor the two expressions)
C)

Question 68

True/False:
If on the interval then must exist

Question 69

Evaluate the limit or state that it does not exist

A)
B)
C)

Question 70

Evaluate the limits using the Squeeze Theorem, trigonometric identities, and trigonometric limits as necessary

A)
B)
C)

Question 71

Compute the following limits:

A)
B)
C)

Question 72

Compute the following limits:

A)
(Hint: Multiply and divide by the conjugate expression.)
B)
(Hint: For .)
C)

Question 73

Compute the following limits:

A)
B)
C)

Question 74

Calculate the limits:

A)
B)
C)

Question 75

Evaluate the limits in terms of the constants involved

A)
B)

Question 76

Show that for all . Use the Squeeze Theorem to evaluate .

Question 77

Find the value of such that the following limit exists, and evaluate the limit for this value:

Question 78

Evaluate the limits

A)
B)
C)

Question 79

Evaluate the limits

A)
B)

Question 80

Find .