Limits and Continuity: Master Calculus Foundations with Interactive Quizzes

Question 1

Calculate the limit of the function (x^2 - 4) / (x - 2) as x approaches 2.

Accepted Answer

0

Answer

1

Answer

2

Answer

4

Question 2

Analyze the continuity of the function f(x) at x = 1:
f(x) = { x^2 - 1, if x < 1
{ 2x - 1, if x >= 1

Accepted Answer

Discontinuous

Answer

Continuous

Answer

Cannot be determined

Question 3

Employ L'Hopital's rule to find the limit of (e^x - 1) / x as x approaches 0.

Accepted Answer

1

Answer

0

Answer

e

Answer

Infinity

Question 4

Which theorem guarantees the existence of a limit?

Accepted Answer

Squeeze Theorem

Answer

Intermediate Value Theorem

Answer

Extreme Value Theorem

Answer

Mean Value Theorem

Question 5

Which of the following is NOT a property of limits?

Accepted Answer

The limit of a function is always unique.

Answer

The limit of a sum is the sum of the limits.

Answer

The limit of a product is the product of the limits.

Answer

The limit of a quotient is the quotient of the limits.

Question 6

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

Accepted Answer

0

Answer

1

Answer

2

Answer

The limit does not exist

Question 7

Is the function f(x) = |x| continuous at x = 0?

Accepted Answer

No, it is not continuous at x = 0.

Answer

Yes, it is continuous at x = 0.

Answer

It is continuous from the left but not from the right at x = 0.

Answer

It is continuous from the right but not from the left at x = 0.

Question 8

Identify the indeterminate form when evaluating the limit of (sin x)/x as x approaches 0.

Accepted Answer

0/0

Answer

infinity/infinity

Answer

1/infinity

Answer

infinity/0

Question 9

Name a technique used to evaluate limits involving indeterminate forms.

Accepted Answer

L'Hopital's Rule

Answer

Squeeze Theorem

Answer

Intermediate Value Theorem

Answer

Substitution

Question 10

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

Accepted Answer

1

Answer

0

Answer

e

Answer

Infinity

Question 11

Which of the following is considered a Cauchy sequence?

Accepted Answer

A sequence whose terms become closer together as the index increases

Answer

A sequence that converges to a limit

Answer

A sequence that is bounded and monotonic

Answer

A sequence whose terms are all identical

Question 12

Evaluate the limit as x approaches 0: (x^2 - 4) / (x - 2)

Accepted Answer

2

Answer

0

Answer

4

Answer

6

Question 13

Determine the continuity of the function f(x) = (x^2 - 1) / (x - 1) at x = 0:

Accepted Answer

Discontinuous

Answer

Continuous

Answer

Cannot be determined

Question 14

Which of the following graphs represents a function that is not continuous at x = 0?

Accepted Answer

A piecewise function with a jump discontinuity at x = 0

Answer

A straight line

Answer

A parabola

Answer

A circle

Question 15

Define the limit of a function f(x) as x approaches a, using the formal mathematical notation:

Accepted Answer

For any positive number (epsilon), there exists a positive number (delta) such that whenever (|x - a| < delta), then (|f(x) - L| < epsilon).

Answer

As x gets closer to a, the value of f(x) gets closer to L.

Answer

The value of f(x) is equal to L when x is equal to a.

Answer

The slope of the tangent line to the graph of f(x) at x = a is equal to L.

Question 16

Which of the following is NOT a property of continuous functions?

Accepted Answer

Continuous functions are always increasing or decreasing.

Answer

The graph of a continuous function has no breaks or jumps.

Answer

Continuous functions can be differentiated everywhere.

Answer

The limit of a continuous function as x approaches a point is equal to the value of the function at that point.

Question 17

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 18

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

Accepted Answer

Yes, it is continuous.

Answer

No, it is not continuous.

Answer

The continuity cannot be determined from the given information.

Question 19

Use the Squeeze Theorem to evaluate lim_(x->0) (x^2 * sin(x)).

Accepted Answer

0

Answer

1

Answer

Does not exist

Answer

Cannot be determined

Question 20

Graph a function that is continuous on the interval (-∞, ∞) except at x = 1, where it has a removable discontinuity.

Accepted Answer

A graph with a hole at x = 1

Answer

A graph with a jump discontinuity at x = 1

Answer

A graph with a vertical asymptote at x = 1

Answer

A graph with an infinite discontinuity at x = 1

Question 21

Find the points of discontinuity of the function k(x) = |x - 2| - 1.

Accepted Answer

x = 2

Answer

x = -2

Answer

x = 0

Answer

The function is continuous at all real numbers

Question 22

Determine if the function f(x) = { x^2 if x ≥ 0, -x^2 if x < 0 } is continuous at x = 0.

Accepted Answer

Continuous at x = 0 because the left and right-hand limits exist and are equal.

Answer

Not continuous at x = 0 because the left and right-hand limits do not exist.

Answer

Not continuous at x = 0 because the function is not defined at x = 0.

Question 23

[Clarity and Precision] Define a limit in the context of functions.

Accepted Answer

The value that a function steadily approaches as the input gets infinitely close to a specific value

Answer

The exact value of a function at a specific input

Answer

The slope of a function at a specific input

Answer

The area under the curve of a function over a specific interval

Question 24

[Language Accessibility] Which of the following statements about the limit of a function as x approaches infinity is correct?

Accepted Answer

It might be infinity, negative infinity, or may not exist

Answer

It must be a finite number

Answer

It must be negative infinity

Answer

It must equal the value of the function at infinity

Question 25

[Language Accessibility] Determine the values of c for which the function f(x) = (x^2 - 4) / (x - c) has a discontinuity.

Accepted Answer

c = 2 and c = 4

Answer

c = 0

Answer

c = 2

Answer

c = 4

Question 26

[Correctness and Precision of the Answer] Which of the following is a consequence of the Intermediate Value Theorem?

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 is differentiable on an interval, its graph cannot have sharp corners

Answer

If a function is integrable on an interval, it has an antiderivative

Answer

If a function is periodic, it has a Fourier series expansion

Question 27

[Correctness and Precision of the Answer] Which of the following functions is continuous but not differentiable at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = ln x

Question 28

Determine whether the function f(x) = x^3 - x is continuous on the interval (-∞, ∞)

Accepted Answer

Continuous

Answer

Discontinuous

Answer

Continuous except at x = √2

Question 29

Determine whether the function f(x) = 1 / (x^2 - 1) is continuous on the interval (-1, 1)

Accepted Answer

Continuous except at x = ±1

Answer

Continuous

Answer

Discontinuous

Question 30

Find the limit as x approaches infinity of the function f(x) = (2x³ - 5x) / (x² + 1).

Accepted Answer

2

Answer

0

Answer

1

Answer

∞

Question 31

Determine whether the function f(x) = |x - 2| is continuous at x = 2.

Accepted Answer

No

Answer

Yes, from the right only

Answer

Yes, at both points

Answer

Yes, from the left only

Question 32

Evaluate the limit: lim_(x->2) (x^2 + 3x - 4) / (x - 2)

Accepted Answer

7

Answer

9

Answer

11

Answer

5

Question 33

Determine whether the function f(x) = |x| / x is continuous at x = 0

Accepted Answer

Discontinuous

Answer

Removable discontinuity

Answer

Continuous

Question 34

Which of the following conditions must be met for a function f(x) to be continuous at a point x = a?

Accepted Answer

f(a) is defined, lim (x->a) f(x) exists, and lim (x->a) f(x) = f(a)

Answer

f(a) exists

Answer

lim (x->a) f(x) exists, and lim (x->a) f(x) = f(a)

Question 35

Find the limit as x approaches infinity of the function f(x) = (3x^2 + 2x) / (x^2 - 1).

Accepted Answer

3

Answer

-1

Answer

0

Answer

Does not exist

Question 36

The limit of a function as x approaches a certain value represents the value the function 'approaches' as x gets closer and closer to that value. This concept is crucial in understanding the behavior of functions at points where they may not be defined. Which of the following is NOT a valid application of limits?

Accepted Answer

Determining the exact value of a function at a specific point.

Answer

Calculating the rate of change of a function at a specific point.

Answer

Analyzing the behavior of a function near a point of discontinuity.

Question 37

The limit of f(x) as x approaches infinity is 3. What can we conclude about the function f(x)?

Accepted Answer

The function has a horizontal asymptote at y = 3

Answer

The function has a vertical asymptote at x = 3

Answer

The function is unbounded as x approaches infinity

Answer

The function is continuous at x = 3

Question 38

What is the value of the limit as x approaches 0 of sin(x) / x?

Accepted Answer

1

Answer

Infinity

Answer

Does not exist

Answer

0

Question 39

Which of the following is a true statement about the function f(x) = |x| at x = 0?

Accepted Answer

The function is continuous but not differentiable at x = 0

Answer

The function is differentiable but not continuous at x = 0

Answer

The function is both continuous and differentiable at x = 0

Question 40

If the limit of f(x) as x approaches a exists, what can we say about the limit of f(x) as x approaches a from the left?

Accepted Answer

It must also exist and be equal to the limit as x approaches a.

Answer

It must be equal to the limit as x approaches a from the right.

Answer

It may or may not exist.

Question 41

A function f(x) is said to be continuous at x = a if:

Accepted Answer

The limit of f(x) as x approaches a exists, f(a) is defined, and the limit equals f(a).

Answer

The limit of f(x) as x approaches a does not exist, but f(a) is defined.

Answer

The limit of f(x) as x approaches a exists, but f(a) is not defined.

Question 42

Find the limit as x approaches 2 of the function f(x) = (x^2 - 4) / (x - 2).

Accepted Answer

4

Answer

Does not exist

Answer

2

Question 43

Which of the following conditions must be met for a function to be continuous at a point x = c?

Accepted Answer

All of the above

Answer

The limit as x approaches c of f(x) must exist.

Answer

f(c) must be defined.

Answer

The limit as x approaches c of f(x) must equal f(c).

Question 44

The function f(x) = 1/x is discontinuous at x = 0. Which type of discontinuity does it have?

Accepted Answer

Infinite Discontinuity

Answer

Removable Discontinuity

Answer

Jump Discontinuity

Answer

Essential Discontinuity

Question 45

If the limit as x approaches 3 of f(x) = 5 and the limit as x approaches 3 of g(x) = 2, what is the limit as x approaches 3 of f(x) + g(x)?

Accepted Answer

7

Answer

3

Answer

10

Answer

Does not exist

Question 46

Consider the function f(x) = {x^2, x < 2; 4, x = 2; 2x + 1, x > 2}. Is this function continuous at x = 2?

Accepted Answer

No

Answer

Yes

Answer

It depends on the value of f(2)

Question 47

What is the limit as x approaches infinity of the function f(x) = (3x^2 + 2x) / (x^2 - 1)?

Accepted Answer

3

Answer

Does not exist

Answer

Infinity

Answer

0

Question 48

Find the value of 'a' that makes the function f(x) = {x^2 - 4, x < 2; ax + 2, x ≥ 2} continuous at x = 2.

Accepted Answer

2

Answer

1

Answer

4

Answer

3

Question 49

Which of the following is NOT a property of limits?

Accepted Answer

The limit of the product of two functions is the product of the limits of each function.

Answer

The limit of a sum of two functions is the sum of the limits of each function.

Answer

The limit of a constant function is the constant.

Question 50

A function f(x) is said to have a removable discontinuity at x = c if...

Accepted Answer

The limit as x approaches c of f(x) exists but is not equal to f(c).

Answer

The function is discontinuous at x = c, but the discontinuity can be removed by redefining f(c).

Answer

The limit as x approaches c of f(x) does not exist.

Question 51

Consider the graph of a function f(x). At x = a, the graph has a vertical asymptote. What does this tell us about the limit of f(x) as x approaches a?

Accepted Answer

The limit does not exist.

Answer

The limit is equal to infinity.

Answer

The limit exists and is finite.

Answer

The limit is equal to zero.