Limits and Continuity: Interactive Learning Quizzes for Calculus I

Question 1

Define continuity at a point c for a function f(x).

Accepted Answer

The function f(x) is defined at c and the limit of f(x) as x approaches c is equal to f(c).

Answer

The limit of f(x) as x approaches c exists.

Answer

The function f(x) is continuous on an open interval containing c.

Answer

The derivative of f(x) exists at c.

Question 2

Which of the following is NOT a type of discontinuity?

Accepted Answer

Continuous discontinuity

Answer

Jump discontinuity

Answer

Infinite discontinuity

Answer

Removable discontinuity

Question 3

Which of the following is a sufficient condition for a function to be continuous on an interval?

Accepted Answer

The function is continuous at every point on the interval.

Answer

The function is differentiable on the interval.

Answer

The function is defined on the interval.

Answer

The function has a finite number of discontinuities on the interval.

Question 4

State the Intermediate Value Theorem.

Accepted Answer

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

Answer

If a function is differentiable on an interval, then it has a minimum and maximum value on the interval.

Answer

If a function is increasing on an interval, then it is continuous on the interval.

Answer

If a function is decreasing on an interval, then it is differentiable on the interval.

Question 5

Which of the following is NOT an application of the Intermediate Value Theorem?

Accepted Answer

Proving that a function is surjective.

Answer

Finding a root of a function.

Answer

Finding the maximum value of a function on an interval.

Answer

Finding the minimum value of a function on an interval.

Question 6

Provide an example of a function that is continuous but not differentiable at a specific point.

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 7

Which property below does NOT apply to continuous functions?

Accepted Answer

Preserves derivatives.

Answer

Preserves inequalities.

Answer

Preserves limits.

Answer

Preserves integrals.

Question 8

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

Accepted Answer

The function is defined and continuous on an open interval containing x.

Answer

The function is differentiable at x.

Answer

The function is piecewise continuous at x.

Answer

The graph of the function has no holes or jumps at x.

Question 9

Let f(x) = (x-1)/(x^2-1). What is the value of f(1)?

Accepted Answer

0

Answer

1/2

Answer

1

Answer

Undefined

Question 10

Which of the following is a property of a continuous function?

Accepted Answer

The graph of the function has no holes or jumps.

Answer

The function is differentiable at every point.

Answer

The function is integrable on any closed interval.

Answer

The function has a maximum and minimum value on any closed interval.

Question 11

Let f(x) = x^2 - 1. What type of discontinuity does f(x) have at x = 1?

Accepted Answer

Jump discontinuity

Answer

Infinite discontinuity

Answer

Removable discontinuity

Answer

Oscillating discontinuity

Question 12

Which of the following accurately defines continuity at a point 'a'?

Accepted Answer

The limit of the function exists at 'a' and equals the function's value at 'a'.

Answer

The function is defined at 'a'.

Answer

The graph of the function has no breaks at 'a'.

Answer

The function is differentiable at 'a'.

Question 13

Which theorem can be employed to demonstrate a function's continuity on an interval?

Accepted Answer

Intermediate Value Theorem

Answer

Extreme Value Theorem

Answer

Rolle's Theorem

Answer

Mean Value Theorem

Question 14

Identify the function that is NOT continuous on the interval (-∞, ∞).

Accepted Answer

f(x) = 1/x

Answer

f(x) = x^2

Answer

f(x) = |x|

Answer

f(x) = sin(x)

Question 15

Provide an application of the Intermediate Value Theorem.

Accepted Answer

Establishing the existence of function roots within an interval

Answer

Determining function maximum and minimum values on a closed interval

Answer

Assessing function differentiability at a specific point

Answer

Calculating the area beneath a curve

Question 16

Which function exhibits discontinuity at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 17

How does the Intermediate Value Theorem help establish solutions to equations in the form f(x) = c?

Accepted Answer

Apply the Intermediate Value Theorem to the interval [a, b] where f(a) < c < f(b)

Answer

Divide the interval into subintervals and apply the definition of continuity

Answer

Find the derivative of f(x) and set it equal to zero

Answer

Cannot be determined

Question 18

In Calculus, how does the continuity of a function affect its integrability?

Accepted Answer

Continuous functions on closed intervals are integrable on those intervals.

Answer

Integrable functions are always continuous.

Answer

Continuity and integrability are independent properties of functions.

Question 19

Which statement accurately describes a function that is continuous on an open interval?

Accepted Answer

It has a limit at every point within the interval.

Answer

It has a finite derivative at every point within the interval.

Answer

It is differentiable at every point within the interval.

Answer

It is differentiable at most points within the interval.

Question 20

Identify the option that does NOT represent a type of discontinuity:

Accepted Answer

Differentiable discontinuity

Answer

Jump discontinuity

Answer

Removable discontinuity

Answer

Infinite discontinuity

Question 21

What primary purpose does the Intermediate Value Theorem serve?

Accepted Answer

Determining if a function has a root on an interval

Answer

Determining if a function is continuous on an interval

Answer

Determining if a function has a derivative at a point

Answer

Determining if a function is differentiable on an interval

Question 22

Which statement accurately describes the graph of a continuous function?

Accepted Answer

It lacks sharp corners or jumps.

Answer

It is always a straight line.

Answer

It is always increasing.

Answer

It is always decreasing.

Question 23

Identify the function that is continuous but not differentiable at a point:

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = 1 / x

Answer

f(x) = e^x

Question 24

Which function exhibits discontinuity at x = 0?

Accepted Answer

f(x) = 1 / x^2

Answer

f(x) = sin(x)

Answer

f(x) = x^3

Answer

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

Question 25

State a consequence of the Intermediate Value Theorem:

Accepted Answer

A continuous function on an interval has a root on the interval.

Answer

A continuous function on an interval is differentiable on the interval.

Answer

A continuous function on an interval is integrable on the interval.

Answer

A continuous function on an interval has a maximum and minimum value on the interval.

Question 26

Which of the following is NOT a necessary condition for a function to be continuous at a point?

Accepted Answer

The graph of the function is continuous at the point.

Answer

The limit of the function as x approaches the point exists.

Answer

The function is defined at the point.

Answer

The value of the function at the point is equal to the limit.

Question 27

Consider the function f(x) = |x-2|. Which of the following statements is true?

Accepted Answer

f(x) has a jump discontinuity at x = 2.

Answer

f(x) is discontinuous at x = 2.

Answer

f(x) has a removable discontinuity at x = 2.

Answer

f(x) has an infinite discontinuity at x = 2.

Question 28

What is the Intermediate Value Theorem?

Accepted Answer

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

Answer

If the limit of a function as x approaches a point exists, then the function is continuous at that point.

Answer

If a function is differentiable on an open interval, then it is continuous on that interval.

Answer

If a function has no discontinuities, then it is continuous on its entire domain.

Question 29

Consider the function f(x) = 1 / (x-2). At which point is the function discontinuous?

Accepted Answer

x = 2

Answer

x = 0

Answer

x = 1

Answer

The function is continuous at all points.

Question 30

Which of the following limits is equal to ∞?

Accepted Answer

lim (x->∞) (e^x) / (x^2 + 1)

Answer

lim (x->∞) (x^2 - 2x + 1) / (x^2 + 1)

Answer

lim (x->∞) (sin(x) + cos(x)) / (x^2 + 1)

Answer

lim (x->∞) (x) / (x^2 + 1)

Question 31

To ensure continuity of a function at a point 'a', which condition must be met?

Accepted Answer

Both B and C

Answer

The function is defined at 'a'.

Answer

The limit of the function as x approaches 'a' exists.

Answer

The value of the function at 'a' is equal to the limit of the function as x approaches 'a'.

Question 32

Which theorem guarantees the existence of a point c in (a, b) such that f(c) = 0, given that f(x) is continuous on [a, b] and f(a) and f(b) have opposite signs?

Accepted Answer

Intermediate Value Theorem

Answer

Extreme Value Theorem

Answer

Mean Value Theorem

Answer

Rolle's Theorem

Question 33

Which of the following functions is continuous over the entire real number line?

Accepted Answer

f(x) = x^3 + 2x^2 - 5

Answer

f(x) = 1/x

Answer

f(x) = |x-1|

Answer

f(x) = tan(x)

Question 34

Regarding the function f(x) = (x-1)/(x-2), which statement is accurate?

Accepted Answer

f(x) has a removable discontinuity at x = 2.

Answer

f(x) is continuous at x = 2.

Answer

f(x) has an infinite discontinuity at x = 2.

Answer

f(x) is discontinuous at every real number.

Question 35

Which of the following limits fails to exist?

Accepted Answer

lim_(x->1) 1/(x-1)

Answer

lim_(x->2) (x^2-4)/(x-2)

Answer

lim_(x->0) sin(x)/x

Answer

lim_(x->infinity) e^x

Question 36

Which consequence is NOT implied by the Intermediate Value Theorem?

Accepted Answer

If f(x) is continuous on [a, b], then there exists c in (a, b) such that f'(c) = 0.

Answer

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

Answer

If f(x) is continuous on [a, b] and f(x) takes every value between f(a) and f(b) for some x in [a, b], then f(x) is constant on [a, b].

Answer

If f(x) is continuous on [a, b] and is increasing on [a, b], then f(x) is differentiable on [a, b].

Question 37

Determine the continuity of the function f(x) = |x| at the point x = 0.

Accepted Answer

Continuous

Answer

Discontinuous from the right

Answer

Continuous from the left, but not from the right

Answer

Discontinuous from the left

Question 38

Determine if the function f(x) = 1 / x is continuous at x = 0.

Accepted Answer

Discontinuous

Answer

Continuous from the left, but not from the right

Answer

Continuous

Answer

Continuous from the right, but not from the left

Question 39

Determine the derivative of the function f(x) = |x - 1| at x = 1.

Accepted Answer

Does not exist

Answer

-1

Answer

0

Answer

1

Question 40

Which of the following functions is continuous at all real numbers?

Accepted Answer

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

Answer

f(x) = |x| / x

Answer

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

Answer

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

Question 41

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 intermediate values between its minimum and maximum values on that interval.

Answer

If a function is continuous on an interval, it is differentiable on that interval.

Answer

If a function is continuous on an interval, it is increasing on that interval.

Question 42

Is the function  *f(x) = (x - 2) / (x² - 4)* continuous at *x = 2*?

Accepted Answer

No, it is discontinuous because the denominator becomes zero at *x = 2*, leading to an undefined value.

Answer

Yes, it is continuous because the function can be simplified by factoring and canceling out the common factor (x - 2).

Answer

Yes, it is continuous because the function approaches a finite value as *x* approaches 2.

Question 43

Using the Intermediate Value Theorem, does the function *f(x) = x³ - 2x + 1* have a root between *x = 0* and *x = 1*?

Accepted Answer

Yes, because *f(0) = 1* and *f(1) = 0*, and *f(x)* is continuous on the interval [0, 1].

Answer

No, because the function is not differentiable on the interval [0, 1].

Answer

No, because the function is not defined at *x = 1*.

Question 44

Is the function *f(x) = 1/x* continuous at *x = 0*?

Accepted Answer

No, it is not continuous because the function is not defined at *x = 0*.

Answer

Yes, it is continuous because the limit as *x* approaches 0 exists and is equal to 0.

Answer

Yes, it is continuous because the function is differentiable everywhere except at *x = 0*.

Question 45

What is the domain on which the function *f(x) = √(x - 1)* is continuous?

Accepted Answer

[1, ∞)

Answer

[-1, ∞)

Answer

(1, ∞]

Question 46

Is the function *f(x) = sin(x) / x* continuous at *x = 0*?

Accepted Answer

Yes, it is continuous because the limit as *x* approaches 0 exists and is equal to 1. We define *f(0) = 1* to make the function continuous at *x = 0*.

Answer

No, it is not continuous because the function is not differentiable at *x = 0*.

Answer

No, it is not continuous because the function is not defined at *x = 0*.

Question 47

Is the function *f(x) = x² + |x|* differentiable at *x = 0*?

Accepted Answer

No, it is not differentiable because the left-hand and right-hand derivatives at *x = 0* are different.

Answer

Yes, it is differentiable because the function is continuous at *x = 0*.

Answer

Yes, it is differentiable because the function is increasing at *x = 0*.

Question 48

Use the definition of continuity to prove that the function *f(x) = x²* is continuous at *x = 1*.

Accepted Answer

For any ε > 0, we can choose δ = √ε such that |f(x) - f(1)| = |x² - 1| = |(x - 1)(x + 1)| ≤ |x - 1| < ε whenever |x - 1| < δ.

Answer

The function is continuous at *x = 1* because it is differentiable at *x = 1*.

Question 49

Is the function *f(x) = {x if x is rational, x² if x is irrational}* continuous at *x = 0*?

Accepted Answer

No, it is not continuous because the left-hand and right-hand limits at *x = 0* are different.

Answer

Yes, it is continuous because the function is defined at *x = 0*.

Answer

Yes, it is continuous because the function is differentiable at *x = 0*.

Question 50

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

Accepted Answer

The limit of the function as *x* approaches *c* exists.

Answer

The function must have a vertical asymptote at *c*.

Answer

The function must be defined at *c*.

Question 51

The function *f(x) = 1/x* has a discontinuity at *x = 0*. Which of the following accurately describes the type of discontinuity?

Accepted Answer

Infinite discontinuity

Answer

Essential discontinuity

Answer

Removable discontinuity

Answer

Jump discontinuity

Question 52

Consider the function *f(x) = (x^2 - 1)/(x - 1)*. Which of the following actions will remove the discontinuity at *x = 1*?

Accepted Answer

Redefining *f(1) = 2*.

Answer

Taking the limit of the function as *x* approaches 1.

Answer

Multiplying the function by (x - 1).

Answer

This discontinuity cannot be removed.

Question 53

According to the Intermediate Value Theorem, which of the following is guaranteed for a continuous function *f(x)* on the interval [-2, 2]?

Accepted Answer

There exists a value *c* in the interval [-2, 2] such that *f(c) = 0*.

Answer

The function is differentiable on the interval [-2, 2].

Answer

The function attains its maximum and minimum values within the interval.

Question 54

Let *f(x) = {x^2, x < 2; 3x - 2, x ≥ 2}* . Determine the continuity of *f(x)* at *x = 2*.

Accepted Answer

Discontinuous because the limits of *f(x)* as *x* approaches 2 from the left and right are not equal.

Answer

Continuous because *f(2)* is defined.

Question 55

Which of the following functions is continuous throughout the real number line?

Accepted Answer

*f(x) = sin(x)*

Answer

*f(x) = |x|*

Answer

*f(x) = tan(x)*

Answer

*f(x) = 1/x*

Question 56

Consider a continuous function *f(x)* on the interval [a, b]. Which of the following statements is false?

Accepted Answer

The function *f(x)* can have a non-removable discontinuity at a point in the interval.

Answer

The function *f(x)* takes on all values between *f(a)* and *f(b)*.

Answer

The function *f(x)* attains its maximum and minimum values on the interval [a, b].

Question 57

The function *f(x) = {x^2, x ≤ 1; 2x, x > 1}* has a discontinuity at *x = 1*. Classify this discontinuity.

Accepted Answer

Jump discontinuity

Answer

Removable discontinuity

Answer

Infinite discontinuity

Answer

Essential discontinuity

Question 58

For the function *f(x) = (x^2 - 9)/(x - 3)* to be continuous at *x = 3*, what value should be assigned to *f(3)*?

Accepted Answer

*f(3) = 6*

Answer

*f(3) = 3*

Answer

*f(3) = 9*

Answer

*f(3) = 0*

Question 59

The function *f(x) = {sin(x), x ≠ 0; 1, x = 0}* is continuous at *x = 0*. What type of discontinuity does this represent?

Accepted Answer

Removable discontinuity

Answer

Essential discontinuity

Answer

Jump discontinuity

Answer

Infinite discontinuity