Limits and Continuity: Master Calculus Concepts with Interactive Quizzes

Question 1

The definition of a limit is:

Accepted Answer

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

Answer

The value that a function equals at a specific point

Answer

The rate of change of a function at a specific point

Answer

The area under a curve at a specific point

Question 2

Which property is NOT true for continuous functions?

Accepted Answer

Derivative is continuous at every point in its domain

Answer

Defined at every point in its domain

Answer

Limit as x approaches any point in its domain equals the value of the function at that point

Answer

Graph has no breaks or jumps

Question 3

Using L'Hopital's rule, evaluate the limit of (sin(x)) / x as x approaches 0:

Accepted Answer

1

Answer

0

Answer

Undefined

Answer

Infinity

Question 4

Which function has an infinite discontinuity at x = 0?

Accepted Answer

f(x) = 1 / x

Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Question 5

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

Accepted Answer

0

Answer

1

Answer

-1

Answer

Does not exist

Question 6

Identify the points where the function f(x) = |x - 1| is discontinuous and explain the reason for the discontinuity.

Accepted Answer

f(x) is discontinuous at x = 1.

Answer

f(x) is continuous at all points.

Answer

f(x) is discontinuous at x = 0.

Answer

f(x) is discontinuous at x = 2.

Question 7

Consider the function h(x) = 1 / (x - 3). Explain why the function is not defined at x = 3 and calculate the limit of h(x) as x approaches 3.

Accepted Answer

Does not exist

Answer

3

Answer

0

Answer

1

Question 8

If f(x) is continuous at x = a, which of the following must be true? Provide a brief explanation.

Accepted Answer

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

Answer

f(a) = 0

Answer

The derivative of f(x) exists at x = a.

Answer

f(x) is differentiable at x = a.

Question 9

Determine the domain of the function f(x) = sqrt(x^2 - 4), taking into account the properties of square roots and any restrictions on the input values.

Accepted Answer

[-2, 2] U [2, infinity)

Answer

(2, infinity)

Answer

[-4, 4]

Answer

[-2, 2]

Question 10

Given that the limit of f(x) as x approaches 'a' is L, and the limit of g(x) as x approaches 'a' is M, what is the limit of [f(x) + g(x)] as x approaches 'a'?

Accepted Answer

L + M

Answer

L - M

Answer

L / M

Answer

L * M

Question 11

At which point is the function f(x) = 1/x discontinuous?

Accepted Answer

x = 0

Answer

The function is continuous everywhere

Answer

x = -1

Answer

x = 1

Question 12

If the limit of f(x) as x approaches 2 is 3, and the limit of g(x) as x approaches 2 is 4, what is the limit of [f(x) * g(x)] as x approaches 2?

Accepted Answer

12

Answer

Does not exist

Answer

1

Answer

7

Question 13

Which of these functions is NOT continuous at x = 1?

Accepted Answer

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

Answer

f(x) = sin(x)

Answer

f(x) = x^2

Answer

f(x) = |x-1|

Question 14

What is the limit of the function f(x) = (x^3 - 8) / (x - 2) as x approaches 2?

Accepted Answer

12

Answer

8

Answer

2

Answer

Does not exist

Question 15

What is the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2?

Accepted Answer

4

Answer

Does not exist

Answer

2

Answer

0

Question 16

What is true about the function f(x) = |x| at x = 0?

Accepted Answer

The function is continuous but not differentiable

Answer

The function is both continuous and differentiable

Answer

The function is neither continuous nor differentiable

Answer

The function is differentiable but not continuous

Question 17

If the limit of f(x) as x approaches 'a' is L, what can we conclude about the value of f(a)?

Accepted Answer

We cannot conclude anything about f(a) based solely on the limit.

Answer

f(a) must be equal to L

Answer

f(a) must be undefined

Question 18

For what value of x is the function f(x) = 1/x discontinuous?

Accepted Answer

x = 0

Answer

x = ∞

Answer

x = -1

Answer

x = 1

Question 19

What is the limit of the function f(x) = sin(x) / x as x approaches 0?

Accepted Answer

1

Answer

Does not exist

Answer

0

Answer

∞

Question 20

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

Accepted Answer

The limit of f(x) as x approaches a exists and is equal to f(a)

Answer

The limit of f(x) as x approaches a exists

Answer

f(a) is defined

Question 21

Given the function f(x) = {x^2, if x < 2; 4, if x ≥ 2}, what is the limit of f(x) as x approaches 2?

Accepted Answer

4

Answer

Does not exist

Answer

2

Question 22

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

Accepted Answer

f(x) = x^3

Answer

f(x) = |x| / x

Answer

f(x) = 1/x

Question 23

If the limit of f(x) as x approaches a from the left is different from the limit of f(x) as x approaches a from the right, what can we conclude?

Accepted Answer

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

Answer

f(x) is continuous at x = a

Answer

f(a) is undefined

Answer

f(x) is differentiable at x = a

Question 24

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

Accepted Answer

The limit of the difference quotient as h approaches 0 exists

Answer

f(x) is continuous at x = a

Answer

The derivative of f(x) at x = a is equal to f(a)