Limits and Continuity: Interactive Quizzes and Practice Questions

Question 1

Determine the vertical asymptote(s) of the function f(x) = (x - 2) / (x^2 - 4).

Accepted Answer

x = 2 and x = -2

Answer

x = 2

Answer

x = -2

Answer

The function has no vertical asymptotes

Question 2

Which of the following theorems is used to prove that continuous functions on closed intervals attain extreme values?

Accepted Answer

Extreme Value Theorem

Answer

Intermediate Value Theorem

Answer

Cauchy-Schwarz Inequality

Answer

Rolle's Theorem

Question 3

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

9

Question 4

Evaluate the limit of the function f(x) = e^x as x approaches positive infinity.

Accepted Answer

∞

Answer

0

Answer

1

Answer

e

Question 5

Find the limit of the function f(x) = sin(x)/x as x approaches 0.

Accepted Answer

0

Answer

1

Answer

π/2

Answer

∞

Question 6

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

Accepted Answer

The left-hand limit and right-hand limit are equal.

Answer

The function is defined at the point.

Answer

The function is differentiable at the point.

Answer

The function is continuous on an interval containing the point.

Question 7

Which of the following functions is not continuous on the interval [-1, 1]?

Accepted Answer

f(x) = 1/x

Answer

f(x) = x^2

Answer

f(x) = |x|

Answer

f(x) = sin(x)

Question 8

Define the limit of a function.

Accepted Answer

The value that the function approaches as the input approaches a specific value

Answer

The value that the function equals at a particular input

Answer

The slope of the function at a specific input

Answer

The area enclosed by the curve of the function between two points

Question 9

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

Accepted Answer

Continuous but not differentiable

Answer

Continuous and differentiable

Answer

Neither continuous nor differentiable

Answer

Differentiable but not continuous

Question 10

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

Accepted Answer

The one-sided limits exist and are equal

Answer

The function is defined at the point

Answer

The derivative exists at the point

Answer

The function is differentiable at the point

Question 11

Find the discontinuity of the function f(x) = sqrt(x - 2):

Accepted Answer

x = 2

Answer

x = -2

Answer

x = 0

Answer

The function has no discontinuities

Question 12

Which of the following is a type of discontinuity where the function has a hole?

Accepted Answer

Removable discontinuity

Answer

Jump discontinuity

Answer

Infinite discontinuity

Answer

Oscillating discontinuity

Question 13

Which of the following properties implies that a function is continuous on an interval?

Accepted Answer

The function is continuous at every point in the interval

Answer

The function is differentiable on the interval

Answer

The function has no zeros on the interval

Answer

The function is bounded on the interval

Question 14

Find the limit of the function f(x) = (1 + x)^1/x as x approaches infinity:

Accepted Answer

e

Answer

0

Answer

1

Answer

The limit does not exist

Question 15

Which of the following is an example of a function that is discontinuous at every rational number?

Accepted Answer

The Dirichlet function

Answer

The floor function

Answer

The ceiling function

Answer

The exponential function

Question 16

What is the best definition of a function's limit at a specific point?

Accepted Answer

The limit is the value the function gets closer to as the input approaches that point.

Answer

The limit is the exact value of the function at that point.

Answer

The limit is always the same as the derivative of the function at that point.

Question 17

Which of the following best defines a limit?

Accepted Answer

The value that a function approaches as the input approaches a specific value

Answer

The value of a function at a specific input

Answer

The derivative of a function at a specific input

Answer

The area under the curve of a function between two inputs

Question 18

What is the negation of the statement "If f(x) is continuous, then f(x) is differentiable"?

Accepted Answer

There exists a function f(x) that is continuous but not differentiable

Answer

If f(x) is differentiable, then f(x) is continuous

Answer

There exists a function f(x) that is differentiable but not continuous

Answer

If f(x) is not continuous, then f(x) is not differentiable

Question 19

Use the Squeeze Theorem to evaluate the limit of sin(x)/x as x approaches 0:

Accepted Answer

0

Answer

1

Answer

The limit does not exist

Answer

Cannot be determined using the Squeeze Theorem

Question 20

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

Accepted Answer

f(x) is continuous but not differentiable at x = 2

Answer

f(x) is differentiable but not continuous at x = 2

Answer

f(x) is both continuous and differentiable at x = 2

Answer

f(x) is neither continuous nor differentiable at x = 2

Question 21

To determine whether a function is continuous at a particular point, specific conditions must be met. Which of the following functions exhibits continuity at x = 2?

Accepted Answer

f(x) = x^2

Answer

f(x) = |x - 2|

Answer

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

Answer

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

Question 22

Regarding the function f(x) = 1/x, which of the following statements accurately describes its properties?

Accepted Answer

Vertical asymptote at x = 0

Answer

Continuous at x = 0

Answer

Differentiable at x = 0

Answer

Horizontal asymptote at y = 0

Question 23

Consider the function f(x) = e^(3x). As x approaches infinity, what is the limit of this function?

Accepted Answer

Infinity

Answer

0

Answer

1

Answer

e

Question 24

Differentiability of a function at a point requires specific conditions. Which of the following is a necessary prerequisite for a function to be differentiable at a given point?

Accepted Answer

Continuity at the point

Answer

Existence of the derivative at the point

Answer

Presence of a vertical tangent at the point

Answer

Presence of a horizontal tangent at the point

Question 25

Determine the equation of the tangent line to the graph of the function f(x) = sin x at x = π/4.

Accepted Answer

y = (√2/2)x + (√2/2)

Answer

y = (√2/2)x - (√2/2)

Answer

y = x + 1

Answer

y = -x + 1

Question 26

Which of the following functions is NOT continuous at x = 1? Explain your answer.

Accepted Answer

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

Answer

f(x) = x^2

Answer

f(x) = |x - 1|

Answer

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

Question 27

Define the precise mathematical definition of a limit.

Accepted Answer

Given any positive number ε, there is a positive number δ such that if the absolute value of (x - a) is less than δ, then the absolute value of (f(x) - L) is less than ε.

Answer

The limit of a function at a point exists if the function is continuous at that point.

Answer

The limit of a function is the value that the function approaches as the input approaches a certain value.

Answer

The limit of a function is the value that the function equals at a certain point.

Question 28

Identify the function that 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 29

Find the horizontal asymptote of the function f(x) = (x^2 + 1) / (x - 1).

Accepted Answer

y = 1

Answer

y = 0

Answer

y = -1

Answer

The function does not have a horizontal asymptote.

Question 30

Identify a vertical asymptote of the function f(x) = 1 / x.

Accepted Answer

x = 0

Answer

x = 1

Answer

x = -1

Answer

The function does not have a vertical asymptote.

Question 31

Determine the limit of the function f(x) = (sin(x) - x) / (x - π/2) as x approaches π/2.

Accepted Answer

0

Answer

1

Answer

-1

Answer

The limit does not exist.

Question 32

The limit of a function f(x) at x = a represents the value that f(x) approaches as x approaches a. Mathematically, we write this as:

Accepted Answer

lim x->a f(x) = L

Answer

lim a->x f(x) = L

Answer

lim f(x)->a x = L

Answer

x->a lim f(x) = L

Question 33

The Intermediate Value Theorem states that:

Accepted Answer

If a function is continuous on an interval and takes on two different values at the endpoints of the interval, it must also take on every value between those two values somewhere in the interval

Answer

If a function is differentiable on an interval, its derivative must be continuous on the interval

Answer

If a function has a limit at a point, then it must be continuous at that point

Answer

If a function is increasing on an interval, then its derivative must be positive on the interval

Question 34

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

Accepted Answer

The function must be differentiable at the point

Answer

The function must be defined at the point

Answer

The limit of the function as x approaches the point must exist

Answer

The value of the function at the point must be equal to the limit

Question 35

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

Accepted Answer

3x^2 + 4x

Answer

3x^2 - 4x

Answer

x^3 + 4x

Answer

x^3 - 4x

Question 36

Define the term 'limit' in mathematics.

Accepted Answer

The value that a function approaches as the input value gets closer and closer to a specific value

Answer

The exact value that a function takes at a specific input

Answer

The rate of change of a function at a specific input

Answer

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

Question 37

Consider the function f(x) = { x^2 if x < 2, 4 if x = 2, x+3 if x > 2 }. At what value of x is f(x) discontinuous?

Accepted Answer

2

Answer

0

Answer

3

Answer

4

Question 38

Evaluate the limit of the sequence {n^2 + 1} / (n^2 - 1) as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

2

Answer

The limit does not exist

Question 39

State a precondition for a function to be continuous at a point.

Accepted Answer

The function's limit at that point exists

Answer

The function is defined at that point

Answer

The function is differentiable at that point

Answer

The function's graph lacks holes at that point

Question 40

Utilize the epsilon-delta definition to demonstrate the continuity of f(x) = x² at x = 2.

Accepted Answer

Given any ε > 0, there exists a δ > 0 such that if |x - 2| < δ, then |f(x) - f(2)| < ε

Answer

Given any ε > 0, there exists a δ > 0 such that if |x - 2| > δ, then |f(x) - f(2)| < ε

Answer

Given any δ > 0, there exists an ε > 0 such that if |x - 2| < δ, then |f(x) - f(2)| > ε

Answer

Given any δ > 0, there exists an ε > 0 such that if |x - 2| > δ, then |f(x) - f(2)| > ε

Question 41

State a direct consequence of the Intermediate Value Theorem.

Accepted Answer

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

Answer

For a continuous function on an interval, its graph is devoid of discontinuities

Answer

For a continuous function on an interval, its derivative is also continuous on that interval

Answer

For a continuous function on an interval, its integral is also continuous on that interval

Question 42

Determine the horizontal asymptotes of f(x) = (x² - 1) / (x - 1).

Accepted Answer

y = 1 and y = -1

Answer

y = 1

Answer

y = -1

Answer

The function has no horizontal asymptotes

Question 43

Which of the following must hold true for a function to be continuous at a point x = a?

Accepted Answer

The function is defined at x = a, the limit of the function as x approaches a exists, and the limit equals the function value at x = a.

Answer

The function is differentiable at x = a.

Question 44

At which point is the function f(x) = {x^2 if x < 2, 4 if x = 2, 2x + 1 if x > 2} discontinuous?

Accepted Answer

x = 2

Answer

x = 1

Answer

The function is continuous at all points.

Answer

x = 0

Question 45

The graph of f(x) has a vertical asymptote at x = 3. What does this indicate about the limit of f(x) as x approaches 3?

Accepted Answer

The limit does not exist.

Answer

The limit is equal to infinity.

Answer

The limit is equal to 3.

Answer

The limit is equal to 0.

Question 46

If lim (x->a) f(x) = L and lim (x->a) g(x) = M, what is lim (x->a) [f(x) + g(x)]?

Accepted Answer

L + M

Answer

L * M

Answer

L - M

Answer

L / M

Question 47

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

Accepted Answer

e

Answer

0

Answer

1

Answer

∞

Question 48

If lim (x->a) f(x) = L and g(x) is continuous at x = a, what is lim (x->a) f(g(x))?

Accepted Answer

f(L)

Answer

L

Answer

g(a)

Answer

g(L)