Limits and Continuity: Master Calculus Fundamentals with Interactive Quizzes

Question 1

Identify a sufficient condition for a function to be continuous at x = a. Explain your choice.

Accepted Answer

The left-hand limit and right-hand limit of the function at x = a are equal.

Answer

The function is defined at x = a.

Answer

The derivative of the function exists at x = a.

Answer

The function is differentiable at x = a.

Question 2

Identify a necessary condition for a function to be differentiable at x = a. Explain your choice.

Accepted Answer

The function is continuous at x = a.

Answer

The derivative of the function exists at x = a.

Answer

The function has a sharp corner at x = a.

Answer

The function has a vertical asymptote at x = a.

Question 3

Identify a graphical representation of a function that is not continuous at x = 2.

Accepted Answer

A graph with a sharp corner at x = 2

Answer

A graph with a smooth curve at x = 2

Answer

A graph with a horizontal asymptote at y = 2

Answer

A graph with a vertical asymptote at x = 3

Question 4

What is the defining characteristic for a function to have a limit at a specific point?

Accepted Answer

The function approaches the same value from both the left and right sides of the point.

Answer

The function approaches different values from the left and right sides of the point.

Answer

The function is continuous at the point.

Question 5

Determine whether the function f(x) = |x - 1| is continuous at x = 1.

Accepted Answer

Discontinuous

Answer

Continuous

Answer

Removable discontinuity

Answer

Infinite discontinuity

Question 6

Which of the following functions is NOT continuous on its entire domain?

Accepted Answer

f(x) = tan(x)

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Answer

f(x) = log(x)

Question 7

Determine the limit of the sequence {n^2 / (n + 1)} as n approaches infinity:

Accepted Answer

1

Answer

0

Answer

Infinity

Answer

The limit does not exist

Question 8

If 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) / 2

Answer

Cannot be determined

Question 9

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

Accepted Answer

The function has a removable discontinuity at the point.

Answer

The function is defined at the point.

Answer

The limit of the function at the point exists.

Answer

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

Question 10

Which of the following statements is true about the limit of sin(x) / x as x approaches 0?

Accepted Answer

The limit is 0.

Answer

The limit is 1.

Answer

The limit does not exist.

Answer

The limit is infinity.

Question 11

Find the limit of the function f(x) = (1 + 1 / x)^x as x approaches infinity. Explain your approach and justify your answer.

Accepted Answer

e

Answer

0

Answer

1

Answer

Does not exist

Question 12

Consider the function g(x) = |x - 2|. Which of the following is true about g(x) at x = 2?

Accepted Answer

g(x) has a jump discontinuity at x = 2

Answer

g(x) is continuous at x = 2

Answer

g(x) has a removable discontinuity at x = 2

Answer

g(x) is discontinuous at every point

Question 13

Consider the function f(x) = x^2 + 2x - 3. Is f(x) continuous for all real numbers?

Accepted Answer

Yes

Answer

No, it is discontinuous at x = -3

Answer

No, it is discontinuous at x = 3

Answer

No, it is discontinuous at both x = -3 and x = 3

Question 14

Which theorem can be used to prove that a function is continuous at a point?

Accepted Answer

Squeeze Theorem

Answer

Chain Rule

Answer

Intermediate Value Theorem

Answer

Mean Value Theorem

Question 15

State the epsilon-delta definition of a limit.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 16

Which function is discontinuous at x = 2?

Accepted Answer

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

Answer

f(x) = x^2 - 1

Answer

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

Answer

f(x) = |x - 2|

Question 17

Find the limit of the sequence a_n = (1 + 1/n)^n as n approaches infinity.

Accepted Answer

e

Answer

0

Answer

1

Answer

∞

Question 18

Which of the following is NOT a theorem related to limits?

Accepted Answer

Fundamental Theorem of Calculus

Answer

Squeeze Theorem

Answer

Sandwich (Intermediate Value) Theorem

Answer

Cauchy's Limit Test

Question 19

Evaluate the limit: lim(x->∞) (x^3 - 2x^2 + 1) / (x^2 + 3x - 1)

Accepted Answer

1

Answer

∞

Answer

-∞

Answer

Does not exist

Question 20

The graph of a function f(x) has a hole at x = 3. What does this imply about the continuity of f(x) at x = 3?

Accepted Answer

f(x) is discontinuous at x = 3.

Answer

f(x) is continuous at x = 3.

Answer

f(x) is differentiable at x = 3.

Answer

f(x) has an infinite discontinuity at x = 3.

Question 21

Find the derivative of the function f(x) = x³ - 2x² + 1.

Accepted Answer

3x² - 4x

Answer

x² - 4x + 1

Answer

3x² - 2x

Answer

x³ - 4x² + 1

Question 22

Find the critical points of the function f(x) = x⁴ - 4x² + 3.

Accepted Answer

x = 0, ±2

Answer

x = ±1, ±2

Answer

x = 0, ±1

Answer

x = ±2

Question 23

Which condition is sufficient for a function to be differentiable?

Accepted Answer

The function has a continuous derivative.

Answer

The function is continuous.

Answer

The function has a finite derivative at every point.

Answer

The function has a local minimum or maximum at every point.

Question 24

What is the limit of the function **f(x) = (x^2-1)/(x-1)** as **x** tends to **1**?

Accepted Answer

2

Answer

Does not exist

Answer

1

Answer

0

Question 25

What is the **vertical asymptote** of the function **f(x) = (x-3)/(x^2 - 9)**?

Accepted Answer

x = 3

Answer

x = -3

Answer

y = -3

Answer

y = 3

Question 26

Which of the following is a graphical representation of a function with a **removable discontinuity** at **x = 2**?

Accepted Answer

A graph with a hole at x = 2.

Answer

A graph that is undefined at x = 2.

Answer

A graph with a jump discontinuity at x = 2.

Answer

A graph with a vertical asymptote at x = 2.

Question 27

What is the derivative of the function **f(x) = ln(x^2 + 1)**?

Accepted Answer

2x/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

(x^2 + 1)/2x

Answer

x^2/(x^2 + 1)

Question 28

Graphically, what does it mean for a function to be continuous at a point?

Accepted Answer

The graph of the function has no sudden jumps or breaks at that point

Answer

The graph of the function has a vertical asymptote at that point

Answer

The function is differentiable at that point

Question 29

Which of the following functions is discontinuous at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = e^x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Question 30

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

Accepted Answer

The function is defined at that point

Answer

The function is differentiable at that point

Answer

The function has a removable discontinuity at that point

Answer

The function has a vertical asymptote at that point

Question 31

Evaluate the limit as x approaches infinity: lim_(x->∞) (x^3 + 2x^2 - 1) / (x^2 + 3x - 2)

Accepted Answer

∞

Answer

-∞

Answer

0

Answer

1

Question 32

Which function is an example of a function that is discontinuous everywhere?

Accepted Answer

f(x) = |x| / x

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = sin(x) + 1

Question 33

What is the limit of the function  f(x) = (x² - 1) / (x - 1) as x approaches 1?

Accepted Answer

2

Answer

0

Answer

Undefined

Answer

-1

Question 34

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

Accepted Answer

1

Answer

0

Answer

Undefined

Answer

Infinity

Question 35

Which graph represents the function f(x) = 1 / x?

Accepted Answer

A hyperbola with vertical asymptote at x = 0 and horizontal asymptote at y = 0

Answer

A line with slope 1

Answer

A circle centered at the origin

Answer

A parabola with vertex at (0, 1)

Question 36

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

Accepted Answer

No, it has a jump discontinuity

Answer

No, it has a removable discontinuity

Answer

No, it has an infinite discontinuity

Answer

Yes

Question 37

Which of these functions is discontinuous at x = 1?

Accepted Answer

f(x) = |x - 1|

Answer

f(x) = e^x

Answer

f(x) = x²

Answer

f(x) = sin(x)

Question 38

Is the function f(x) = { x² if x < 0, 1 if x = 0, x - 1 if x > 0 } continuous at x = 0?

Accepted Answer

No

Answer

Yes

Answer

Depends on the value of 1

Answer

Cannot be determined

Question 39

Which of the following statements is true about the function f(x) = 1 / (x - 2)?

Accepted Answer

It has a vertical asymptote at x = 2.

Answer

It has a horizontal asymptote at y = 0.

Answer

It is differentiable at x = 2.

Answer

It is continuous at x = 2.

Question 40

Which function is not continuous at every real number?

Accepted Answer

f(x) = tan(x)

Answer

f(x) = x²

Answer

f(x) = e^x

Answer

f(x) = cos(x)

Question 41

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

Accepted Answer

4

Answer

0

Answer

2

Answer

Does not exist

Question 42

Given the function f(x) = {x^2, x < 2; 4, x = 2; x + 2, x > 2}, is f(x) continuous at x = 2?

Accepted Answer

No

Answer

Only continuous from the left

Answer

Yes

Answer

Cannot be determined

Question 43

The graph of a function has a vertical asymptote at x = 3. What does this imply about the limit of the function as x approaches 3?

Accepted Answer

The limit does not exist.

Answer

The limit is equal to infinity.

Answer

The limit is equal to negative infinity.

Answer

The limit is equal to 3.

Question 44

Which of the following functions is continuous at x = 1?

Accepted Answer

f(x) = x^2 - 1

Answer

f(x) = |x - 1|

Answer

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

Answer

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

Question 45

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

Accepted Answer

L + M

Answer

L - M

Answer

L * M

Answer

L / M

Question 46

Find the value of k that makes the function f(x) = {2x + 1, x < 1; kx, x ≥ 1} continuous at x = 1.

Accepted Answer

3

Answer

2

Answer

1

Question 47

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

Infinity

Question 48

Which of the following statements is true about the function f(x) = 1/x?

Accepted Answer

The function has a vertical asymptote at x = 0.

Answer

The function is continuous at x = 0.

Answer

The function has a removable discontinuity at x = 0.

Answer

The function has a horizontal asymptote at y = 0.

Question 49

If lim_(x->a) f(x) = L and lim_(x->a) g(x) = M, then what is lim_(x->a) [f(x) * g(x)]?

Accepted Answer

L * M

Answer

L / M

Answer

L + M

Answer

L - M

Question 50

The function f(x) = {x^2, x < 1; 2x, x ≥ 1} is:

Accepted Answer

Continuous at x = 1

Answer

Continuous only from the left at x = 1

Answer

Discontinuous at x = 1

Answer

Continuous only from the right at x = 1