Limits and Continuity: Master Calculus Concepts with Interactive Quizzes

Question 1

What is the formal definition of a limit of a function f(x) as x approaches a specific value c?

Accepted Answer

The value that f(x) approaches as x gets arbitrarily close to c

Answer

The value that f(x) equals when x equals c

Answer

The slope of the tangent line to the graph of f(x) at x = c

Answer

The area under the curve of f(x) between x = c and x = c+1

Question 2

Which of the following is a necessary and sufficient condition for a function to be continuous at a point?

Accepted Answer

The limit of the function exists at that point, and the function is defined at that point.

Answer

The function is differentiable at that point.

Answer

The graph of the function has no breaks at that point.

Answer

The function is continuous on the entire real line.

Question 3

Consider the piecewise-defined function:
f(x) = { x^2, if x > 2
{ 2x, if x <= 2. 
What is the limit of f(x) as x approaches 2 from the left?

Accepted Answer

2

Answer

0

Answer

4

Answer

The limit does not exist.

Question 4

Which of the following properties of continuous functions is false?

Accepted Answer

Continuous functions are differentiable everywhere.

Answer

Continuous functions can be represented by power series.

Answer

Continuous functions are bounded on closed intervals.

Answer

Continuous functions have graphs without breaks or jumps.

Question 5

The definition of a limit is:

Accepted Answer

The value that a function approaches as its input approaches a certain value

Answer

The value that a function equals at a specific input

Answer

The rate of change of a function at a particular point

Answer

The area under the curve of a function between two points

Question 6

A sufficient condition for a function to be continuous at a point is:

Accepted Answer

The function is continuous from the left and right at that point

Answer

The function is differentiable at that point

Answer

The function has a finite value at that point

Answer

The graph of the function has no breaks at that point

Question 7

Which function has a removable discontinuity at x = 1?

Accepted Answer

f(x) = (x^2 - 1)/(x - 1)

Answer

f(x) = (x - 1)/(x - 2)

Answer

f(x) = |x - 1|

Answer

f(x) = e^(x - 1)

Question 8

Which function is continuous everywhere?

Accepted Answer

f(x) = e^x

Answer

f(x) = 1/x

Answer

f(x) = tan(x)

Answer

f(x) = |sin(x)|

Question 9

A necessary condition for a function to be differentiable at a point is:

Accepted Answer

The function is continuous at that point

Answer

The function has a non-zero derivative at that point

Answer

The graph of the function has a sharp corner at that point

Answer

The function is increasing at that point

Question 10

Which function has a vertical asymptote at x = 2?

Accepted Answer

f(x) = (x + 2)/(x - 2)

Answer

f(x) = (x^2 + 4)/(x - 2)

Answer

f(x) = |x - 2|

Answer

f(x) = e^(x - 2)

Question 11

Define the limit of a function.

Accepted Answer

The value a function approaches as its input approaches a certain value

Answer

The value of a function at a particular input

Answer

The slope of a function at a specific point

Answer

The rate of change of a function at a specific point

Question 12

State the necessary condition for a function to be differentiable at a point.

Accepted Answer

Continuity of the function at that point

Answer

Existence of the derivative of the function at that point

Answer

Presence of a vertical tangent line at that point

Answer

Presence of a horizontal tangent line at that point

Question 13

Define the Intermediate Value Theorem.

Accepted Answer

For a continuous function on an interval, it takes on every value between its minimum and maximum values on that interval

Answer

A differentiable function on an interval is also continuous on that interval

Answer

A function with a limit at a point is continuous at that point

Answer

A function with a derivative at a point is differentiable at that point

Question 14

Which function is continuous on the interval [-1, 1] but not differentiable at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = 1 / x

Answer

f(x) = sin(x)

Question 15

Which of the following best defines the limit of a function f(x) as x approaches c?

Accepted Answer

The value that f(x) gets infinitely close to as x gets arbitrarily close to c

Answer

The value of f(c)

Answer

The slope of the tangent line to the graph of f(x) at x = c

Answer

The area under the curve of f(x) from x = c to x = c + 1

Question 16

Which of the following is a graphical representation of a continuous function?

Accepted Answer

A curve that has no sudden breaks or jumps

Answer

A curve that is always increasing or decreasing

Answer

A curve that has a sharp point at every integer value of x

Answer

A curve that oscillates wildly

Question 17

The value of the limit as x approaches infinity of (1 + 1/x)^x is:

Accepted Answer

e

Answer

0

Answer

1

Answer

Infinity

Question 18

Which of the following functions is NOT differentiable at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 19

Regarding the function f(x) = 1 / x at x = 0, which statement is true?

Accepted Answer

f(x) is discontinuous at x = 0

Answer

f(x) is continuous at x = 0

Answer

f(x) has a removable discontinuity at x = 0

Answer

f(x) has an infinite discontinuity at x = 0

Question 20

Which of the following functions is not continuous at x = 0?

Accepted Answer

g(x) = 1/x

Answer

f(x) = x^2

Answer

h(x) = |x|

Answer

k(x) = e^x

Question 21

Find the horizontal asymptote of the function $f(x) = \frac{x^2 + 2x}{x^2 - 1}$.

Accepted Answer

y = 1

Answer

y = 0

Answer

y = 2

Answer

The function does not have a horizontal asymptote.

Question 22

Determine if the function $f(x) = \frac{1}{x - 1}$ is continuous on the interval $(0, 1)$.

Accepted Answer

No

Answer

Yes

Answer

Cannot be determined.

Question 23

Find the limit of the function $f(x) = \frac{2x^2 + 3x - 5}{x^2 - 1}$ as $x$ approaches infinity.

Accepted Answer

5

Answer

2

Answer

3

Answer

The limit does not exist.

Question 24

Find the derivative of the function f(x) = e^(2x + 1).

Accepted Answer

2e^(2x + 1)

Answer

e^(2x + 1)

Answer

2

Answer

1

Question 25

Determine whether the function f(x) = |x| has a maximum or a minimum at the point x = 0.

Accepted Answer

Neither a maximum nor a minimum

Answer

Maximum

Answer

Minimum

Answer

Cannot be determined from the given information

Question 26

Use the Intermediate Value Theorem to prove that the function f(x) = x^3 - 2x^2 + 1 has at least one root in the interval [0, 1].

Accepted Answer

The function is continuous on [0, 1] and f(0) < 0 while f(1) > 0.

Answer

The function is differentiable on [0, 1] and f'(x) > 0 for all x in [0, 1].

Answer

The function is increasing on [0, 1] and f(0) < 0 while f(1) > 0.

Answer

The function is decreasing on [0, 1] and f(0) < 0 while f(1) > 0.

Question 27

Which of the following formally defines the limit of a function f(x) as x approaches a?

Accepted Answer

The value that the function f(x) approaches as x gets closer and closer to a.

Answer

The value of the function f(x) at x = a.

Answer

The slope of the function f(x) at x = a.

Answer

The area under the curve of the function f(x) from x = 0 to x = a.

Question 28

Find the limit of the function f(x) = (x^2 - 9) / (x - 3) as x approaches 3.

Accepted Answer

0

Answer

3

Answer

6

Answer

DNE

Question 29

Find the values of c for which the function f(x) = { x^2 - 4, x ≤ c; 3x - 2, x > c } is continuous at x = c.

Accepted Answer

c = 2

Answer

c = -2

Answer

c = 0

Answer

There are no such values of c.

Question 30

Find the limit of the function f(x) = (e^x - 1) / x as x approaches 0.

Accepted Answer

1

Answer

0

Answer

e

Answer

DNE

Question 31

Which of the following statements about the Intermediate Value Theorem is true?

Accepted Answer

If a function is continuous on a closed interval, then it takes on every value between its minimum and maximum values.

Answer

The Intermediate Value Theorem only applies to functions that are differentiable.

Answer

The Intermediate Value Theorem can be used to find the roots of a function.

Answer

The Intermediate Value Theorem is only true for functions that are defined on the entire real line.

Question 32

Determine if the function f(x) = e^x is continuous on the interval [-1, 2].

Accepted Answer

Continuous

Answer

Discontinuous

Answer

Cannot be determined

Question 33

Which of the following is NOT a theorem that can be used to establish continuity?

Accepted Answer

Chain Rule

Answer

Intermediate Value Theorem

Answer

Squeeze Theorem

Answer

Extreme Value Theorem

Question 34

Determine if the function f(x) = |x| + |x - 1| is differentiable at x = 0.

Accepted Answer

Not differentiable

Answer

Differentiable

Answer

Cannot be determined

Question 35

Identify the theorem that states: If f(x) is continuous on [a, b] and f(a) > 0 and f(b) < 0, then there exists a number c in (a, b) such that f(c) = 0.

Accepted Answer

Intermediate Value Theorem

Answer

Extreme Value Theorem

Answer

Rolle's Theorem

Answer

Mean Value Theorem

Question 36

Find the point(s) of discontinuity of the function f(x) = 1 / (x^2 - 1).

Accepted Answer

x = ±1

Answer

x = 0

Answer

x = 2

Answer

The function is continuous everywhere

Question 37

Determine the derivative of the function f(x) = |x|.

Accepted Answer

1, x > 0; -1, x < 0

Answer

|1|, x ≥ 0; |-1|, x < 0

Answer

1, x ≥ 0; undefined, x < 0

Answer

0

Question 38

Evaluate whether the function f(x) = |x| is continuous at x = 0.

Accepted Answer

Discontinuous

Answer

Continuous

Answer

Continuous from the left only

Answer

Continuous from the right only

Question 39

Find the limit of the function f(x) = sqrt(x + 2) as x approaches infinity.

Accepted Answer

∞

Answer

0

Answer

1

Answer

Limit does not exist

Question 40

The limit of the function f(x) = 1 / (x - 2) as x approaches 2 from the left is -∞

Accepted Answer

-∞

Answer

∞

Answer

0

Answer

2

Question 41

The function f(x) = { x^2 if x < 2, 4 if x = 2, x + 2 if x > 2 } has a jump discontinuity at x = 2

Accepted Answer

No, because the function has a jump discontinuity at x = 2

Answer

Yes, because the function is defined at x = 2 and the limit exists

Answer

No, because the function is not differentiable at x = 2

Answer

Yes, because the function is continuous at all points

Question 42

A necessary condition for a function to be differentiable at a point is that the function is continuous at that point

Accepted Answer

The function is continuous at that point

Answer

The function has a vertical asymptote at that point

Answer

The function has a horizontal asymptote at that point

Answer

The function has a removable discontinuity at that point

Question 43

The limit of the function f(x) = (x^2 - 1) / (x - 1) as x approaches 1 is 2

Accepted Answer

2

Answer

0

Answer

1

Answer

The limit does not exist

Question 44

Precise Definition of Limit: What is the definition of the limit of a function as the input approaches a specific value?

Accepted Answer

The value that the function approaches as the input gets arbitrarily close to the specified value

Answer

The exact value that the function reaches at that input value

Answer

The rate of change of the function at that input value

Answer

The integral of the function over a small interval containing that input value

Question 45

Definition of Continuity over an Interval: When is a function said to be continuous over an interval?

Accepted Answer

It is continuous at every point within the interval

Answer

It is defined at every point within the interval

Answer

Its derivative exists at every point within the interval

Answer

Its graph has no sharp turns within the interval

Question 46

Statement of the Intermediate Value Theorem: What does the Intermediate Value Theorem state?

Accepted Answer

If a function is continuous on an interval, it takes on all values between its minimum and maximum values on that interval

Answer

If a function has a continuous derivative on an interval, its graph has no local maxima or minima

Answer

If a function is increasing on an interval, its derivative is positive on that interval

Answer

If a function is concave up on an interval, its second derivative is positive on that interval

Question 47

Derivative Calculation: Find the derivative of the function f(x) = x^3 + 2x^2 - 5x + 1.

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

3x^2 - 4x - 5

Answer

x^2 - 4x - 5

Question 48

Which of the following functions is not continuous on the interval [0, 1]? Identify the discontinuity and explain why it occurs.

Accepted Answer

f(x) = 1/x

Answer

f(x) = sin(x)

Answer

f(x) = |x|

Answer

f(x) = x^2

Question 49

Use the Squeeze Theorem to find the limit of the function f(x) = x^2 * sin(1/x) as x approaches 0. Explain your steps clearly.

Accepted Answer

0

Answer

1

Answer

Cannot be determined

Answer

Does not exist

Question 50

Find the horizontal asymptote of the function f(x) = (2x^2 + 3x - 5) / (x^2 - 1). Explain how you determined the asymptote.

Accepted Answer

y = 2

Answer

y = -2

Answer

y = 3

Question 51

Which of the following statements about the function f(x) = {1, x = 0}, {0, x ≠ 0} is true? Justify your answer.

Accepted Answer

f(x) is continuous at x = 0 but not differentiable at x = 0.

Answer

f(x) is not continuous at x = 0 but differentiable at x = 0.

Answer

f(x) is both continuous and differentiable at x = 0.

Question 52

Determine whether the function f(x) = e^(x^2) is continuous on the interval (-∞, ∞). Explain your reasoning.

Accepted Answer

Yes

Answer

Continuous on (0, ∞) only

Answer

No

Answer

Continuous on (-∞, 0) only

Question 53

State the necessary condition for a function to be continuous at a point. Explain why this condition is essential for continuity.

Accepted Answer

The limit of the function exists at that point.

Answer

The function is increasing at that point.

Answer

The function is differentiable at that point.

Answer

The function is decreasing at that point.