Calculus Limits: Master the Basics with Interactive Quizzes

Question 1

Continuity Theorem: Which theorem is used to determine the continuity of a function at a point?

Accepted Answer

Limit Definition of Continuity Theorem

Answer

Chain Rule Theorem

Answer

Mean Value Theorem

Answer

Extreme Value Theorem

Question 2

Non-Discontinuity Type: Which of the following is NOT a type of discontinuity?

Accepted Answer

Oscillatory discontinuity

Answer

Removable discontinuity

Answer

Jump discontinuity

Answer

Infinite discontinuity

Question 3

Which theorem establishes that the limit of a continuous function on a closed interval exists?

Accepted Answer

Extreme Value Theorem

Answer

Intermediate Value Theorem

Answer

Cauchy's Mean Value Theorem

Answer

Squeeze Theorem

Question 4

Calculate the limit of the function f(x) = (e^x + x^2)/(e^x - x^2) as x approaches infinity.

Accepted Answer

1

Answer

0

Answer

∞

Answer

-∞

Question 5

Find the limit: lim_(x->0) (sin(x))/x

Accepted Answer

1

Answer

0

Answer

Does not exist

Answer

π

Question 6

Use the limit definition of the derivative to find the derivative of f(x) = x^3.

Accepted Answer

3x^2

Answer

x^2

Answer

0

Answer

3

Question 7

Find the limit: lim_(x->a) sqrt(x - a)

Accepted Answer

sqrt(a)

Answer

0

Answer

a

Answer

∞

Question 8

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

Accepted Answer

x = 1

Answer

x = -1

Answer

x = 0

Answer

The function does not have any vertical asymptotes.

Question 9

Determine if the function f(x) = x^3 - 2x^2 + x - 2 is differentiable at x = 1.

Accepted Answer

Yes, the function is differentiable at x = 1.

Answer

No, the function is not differentiable at x = 1 because the derivative does not exist.

Answer

No, the function is not differentiable at x = 1 because the limit of the derivative does not exist.

Answer

No, the function is not differentiable at x = 1 because the second derivative does not exist.

Question 10

Apply the Squeeze Theorem to find the limit: lim (x^2 sin(x)) / x as x approaches 0.

Accepted Answer

0

Answer

1

Answer

Does not exist

Answer

Cannot be determined from the given information

Question 11

Determine the value of c that makes the function f(x) = (x^2 - 4) / (x - c) continuous at x = c.

Accepted Answer

c = 2

Answer

c = -2

Answer

c = 0

Answer

c = 1

Question 12

Evaluate the limit of the function f(x) = (ln(x) - 1) / x as x approaches infinity.

Accepted Answer

0

Answer

1

Answer

e

Answer

The limit does not exist.

Question 13

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

Accepted Answer

f(x) = x^2 + 1

Answer

f(x) = 1/x

Answer

f(x) = |x|

Answer

f(x) = tan x

Question 14

Which of the following is the definition of a limit?

Accepted Answer

The value that a function approaches as the input approaches some value.

Answer

The value that a function takes on 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.

Question 15

The limit of the function f(x) = (sinx)/x as x approaches 0 is:

Accepted Answer

0

Answer

1

Answer

Does not exist

Answer

Cannot be determined

Question 16

Precise Definition of a Limit: Which statement accurately describes the definition of a limit?

Accepted Answer

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

Answer

The value a function equals at a specific value

Answer

The rate of change of a function at a specific value

Answer

The area under the curve of a function at a specific value

Question 17

Define the limit of a function f(x) as x approaches a specific value c.

Accepted Answer

The value that f(x) gets arbitrarily close to as x gets arbitrarily close to c, if such a value exists.

Answer

The derivative of f(x) at x = c.

Answer

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

Question 18

Which of the following conditions implies that the limit of f(x) as x approaches a does not exist?

Accepted Answer

lim f(x) from the left of a and lim f(x) from the right of a are different.

Answer

lim f(x) = ∞

Answer

f(x) is undefined at x = a.

Question 19

Which of the following methods can be used to evaluate the limit of (x^2 + 1)^x as x approaches infinity?

Accepted Answer

Natural Logarithm

Answer

Squeeze Theorem

Answer

L'Hôpital's Rule

Question 20

Evaluate the limit of the expression (x^2 - 1) / (x - 1) using L'Hôpital's Rule as x approaches 1.

Accepted Answer

2

Answer

∞

Answer

0

Answer

1

Question 21

Define the concept of a limit.

Accepted Answer

The specific value that a function approaches as its input gets arbitrarily close to a particular point.

Answer

The rate of change of a function at a specific input.

Answer

The actual value of a function at a particular input.

Question 22

State a sufficient condition for a function to be continuous at a specific point.

Accepted Answer

The limit of the function at that point must exist and be equal to the value of the function at that point.

Answer

The derivative of the function must exist at that point.

Question 23

Identify an example of a discontinuous function.

Accepted Answer

f(x) = 1 / x at x = 0

Answer

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

Answer

f(x) = sin(x) at x = π/2

Answer

f(x) = x^2 at x = 2

Question 24

Given the function g(x) = (x^2 - 1) / |x - 1|, find the limit of g(x) as x approaches 1 from the left.

Accepted Answer

-2

Answer

2

Answer

1

Answer

0

Question 25

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

Accepted Answer

4

Answer

Undefined

Answer

2

Answer

0

Question 26

Determine the limit of the function g(x) = (sin(x) - x)/x^3 as x approaches 0.

Accepted Answer

-1/6

Answer

0

Answer

1/6

Answer

Undefined

Question 27

Which of the following is the correct definition of a limit?

Accepted Answer

For any positive number ε, there exists a positive number δ such that |f(x) - L| < ε whenever 0 < |x - a| < δ.

Answer

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

Question 28

Use the Squeeze Theorem to determine the limit of the function g(x) = (x^2 - 1)/(x - 1) as x approaches 1.

Accepted Answer

2

Answer

Undefined

Answer

0

Answer

1

Question 29

Is the function f(x) = (x + 2)/(x^2 - 4) continuous at x = 2?

Accepted Answer

No

Answer

Yes

Answer

The function is not defined at x = 2.

Answer

Cannot be determined

Question 30

Calculate the limit of the function h(x) = (sqrt(x^2 + 1) - x)/x as x approaches infinity.

Accepted Answer

1/2

Answer

1

Answer

Infinity

Answer

0

Question 31

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

Accepted Answer

2

Answer

The limit does not exist.

Answer

1

Answer

0

Question 32

Is the following statement true or false: The limit of a function as x approaches a point can exist even if the function is not defined at that point.

Accepted Answer

True

Answer

False

Question 33

If the limit of f(x) as x approaches a is L and the limit of g(x) as x approaches a is M, then the limit of (f(x) / g(x)) as x approaches a is equal to...

Accepted Answer

L/M, if M is not equal to 0

Answer

L - M

Answer

L + M

Answer

L * M

Question 34

At which value of x is the function f(x) = (x^2 - 4)/(x - 2) discontinuous?

Accepted Answer

x = 2

Answer

x = -2

Answer

x = 0

Answer

The function is continuous everywhere.

Question 35

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

Accepted Answer

3

Answer

1

Answer

2

Answer

The limit does not exist.

Question 36

A function is considered continuous at a point if...

Accepted Answer

The limit of the function as x approaches that point exists and is equal to the function's value at that point.

Answer

The function is defined at that point.

Question 37

Which statement about the limit of a function is correct?

Accepted Answer

A function can have a limit at a point even if it is not defined at that point.

Answer

If the limit of a function at a point exists, then the function must be defined at that point.

Answer

If a function is defined at a point, then the limit at that point must exist.

Question 38

Is the following statement true or false: The limit of a function as x approaches a point can be obtained simply by substituting the value of x into the function.

Accepted Answer

False

Answer

True