Limits and Continuity: Master Calculus Fundamentals

Question 1

Find the value of c that makes the function f(x) = { x² + 1, x ≤ 2; cx + 3, x > 2 } continuous at x = 2.

Accepted Answer

c = 7

Answer

c = 5

Answer

c = 9

Answer

c = 11

Question 2

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

Accepted Answer

f(x) has a removable discontinuity at x = 1.

Answer

f(x) has an infinite discontinuity at x = 1.

Answer

f(x) is continuous at x = 1.

Answer

f(x) is not defined at x = 1.

Question 3

Determine whether the function f(x) = arctan x is continuous at all real numbers.

Accepted Answer

Continuous on the interval (-∞, ∞)

Answer

Continuous

Answer

Discontinuous

Answer

Continuous on the interval (-π/2, π/2)

Question 4

Which of the following functions is NOT continuous at x = 0?

Accepted Answer

h(x) = 1 / x

Answer

f(x) = x²

Answer

g(x) = |x|

Answer

k(x) = sin(x)

Question 5

Determine if the function g(x) = { 0 if x is rational, 1 if x is irrational } is continuous at any point.

Accepted Answer

Discontinuous at all points

Answer

Continuous at all points

Answer

Continuous at rational points only

Answer

Continuous at irrational points only

Question 6

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

Accepted Answer

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

Answer

The function must be defined at the point.

Answer

The function must be differentiable at the point.

Answer

The function must be increasing at the point.

Question 7

Which of the following is the definition of continuity at a point c?

Accepted Answer

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

Answer

The function is defined at c.

Answer

The function is differentiable at c.

Answer

The function has no vertical asymptotes at c.

Question 8

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

Accepted Answer

f(x) = |x|

Answer

g(x) = 1 / x

Answer

h(x) = x^2 - 1

Answer

j(x) = tan(x)

Question 9

Which of the following statements is true about the graph of a continuous function?

Accepted Answer

It has no sharp corners or breaks.

Answer

It is always increasing or decreasing.

Answer

It has a slope of 0 at every point.

Answer

It passes the vertical line test.

Question 10

Which of the following functions is not differentiable at x = 0?

Accepted Answer

f(x) = |x|

Answer

g(x) = x^2

Answer

h(x) = sin(x)

Answer

j(x) = e^x

Question 11

What is a vertical asymptote of the function f(x) = 1 / (x - 3)?

Accepted Answer

x = 3

Answer

x = 0

Answer

x = 2

Answer

x = 4

Question 12

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

Accepted Answer

y = 2

Answer

y = -2

Answer

y = 0

Answer

The function does not have a horizontal asymptote.

Question 13

Which of the following is a removable discontinuity of the function f(x) = (x^2 - 9) / (x - 3)?

Accepted Answer

x = 3

Answer

x = 0

Answer

x = 2

Answer

x = 4

Question 14

What are the points of discontinuity for the function f(x) = |x - 2| / (x - 2)?

Accepted Answer

x = 2

Answer

x = 0

Answer

x = 1

Answer

x = 3

Question 15

Identify the discontinuity of the function f(x) = 1 / (x - 3).

Accepted Answer

x = 3

Answer

x = -3

Answer

x = 0

Answer

The function is continuous everywhere

Question 16

Which of the following is not a type of discontinuity?

Accepted Answer

Continuous

Answer

Removable

Answer

Jump

Answer

Infinite

Question 17

Does the function f(x) = sqrt(x - 1) have continuity on the interval [1, infinity)?

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Question 18

Determine if the function f(x) = x^2 - 4 is continuous at x = 2.

Accepted Answer

Continuous

Answer

Discontinuous

Answer

Cannot be determined

Question 19

Evaluate the limit: lim ((x^2 - 4x + 3) / (x - 3)).

Accepted Answer

1

Answer

3

Answer

5

Answer

Does not exist

Question 20

Find the point of discontinuity for the function f(x) = { x^2 if x < 1, 2x - 1 if x ≥ 1 }.

Accepted Answer

x = 1

Answer

x = 0

Answer

x = 2

Answer

No points of discontinuity

Question 21

Determine if the function f(x) = x³ - 8 is continuous at x = 2.

Accepted Answer

Continuous

Answer

Discontinuous - Jump

Answer

Discontinuous - Removable

Answer

Discontinuous - Infinite

Question 22

Determine if the function f(x) = x² - 4x + 4 is continuous on the interval (-∞, ∞).

Accepted Answer

Continuous

Answer

Discontinuous at x = 0

Answer

Discontinuous at x = 2

Answer

Discontinuous at both x = 0 and x = 2

Question 23

The function f(x) = sqrt(x^2 - 4) is continuous but not differentiable at x = 2. Which of the following statements is true?

Accepted Answer

The function is not differentiable at x = 2.

Answer

The function is differentiable at x = 2.

Answer

The function is differentiable for all real values of x.

Answer

The function is not differentiable because it is not continuous at x = 2.

Question 24

Find the limit of the function f(x) = (x² - 4) / (x - 2) as x approaches 2.

Accepted Answer

2

Answer

4

Answer

6

Answer

-2

Question 25

Which of the following statements about the function f(x) = 1 / (x² - 1) is true?

Accepted Answer

Discontinuous at x = 1 and x = -1

Answer

Continuous for all x

Answer

Discontinuous at x = 0

Answer

Discontinuous at x = 1 only

Question 26

Find the limit of the function f(x) = (√(x + 1) - 1) / (x - 1) as x approaches 1.

Accepted Answer

1/2

Answer

0

Answer

1

Answer

Does not exist

Question 27

Which of the following functions has a removable discontinuity at x = 0?

Accepted Answer

f(x) = x / |x|

Answer

f(x) = 1 / x

Answer

f(x) = e^x

Answer

f(x) = sin(1 / x)

Question 28

Determine whether the function g(x) = 1 / (x^2 - 1) has a removable discontinuity at x = 1.

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined without more information

Answer

The function is continuous at x = 1

Question 29

Determine whether the function h(x) = |x - 1| + |x + 1| is differentiable at x = 0.

Accepted Answer

The function has a corner point at x = 0

Answer

Differentiable

Answer

Not differentiable

Answer

Cannot be determined without more information

Question 30

Determine if the function f(x) = |x| is continuous at x = 0. Explain your reasoning.

Accepted Answer

Yes, it is continuous at x = 0.

Answer

No, it is not continuous at x = 0.

Answer

It is continuous for all real numbers.

Answer

It is not continuous at any real number.

Question 31

Given the function f(x) = x^3 - 8, find the limit as x approaches infinity. Justify your answer.

Accepted Answer

Infinity

Answer

0

Answer

8

Answer

-Infinity

Question 32

Suppose f(x) is a polynomial function. Which of the following statements must be true? Explain your choice.

Accepted Answer

f(x) is continuous at all real numbers.

Answer

f(x) has a limit at infinity.

Answer

f(x) is differentiable at all real numbers.

Answer

The graph of f(x) has no sharp corners or cusps.

Question 33

Find the limit of the function f(x) = (e^x - 1) / x as x approaches 0. Show your work.

Accepted Answer

1

Answer

0

Answer

e

Answer

The limit does not exist

Question 34

Find the limit of (e^x - 1) / x as x approaches ∞.

Accepted Answer

1

Answer

0

Answer

e

Answer

∞

Question 35

Determine if the function f(x) = (x³ - 8) / (x - 2) is continuous at x = 2.

Accepted Answer

Yes, since the limit exists and is equal to the value of the function at x = 2.

Answer

No, since the limit does not exist.

Answer

Yes, since the function is defined at x = 2.

Answer

No, since the function has an infinite discontinuity at x = 2.

Question 36

Which of the following is the correct definition of the limit of a function f(x) as x approaches a?

Accepted Answer

The precise value that f(x) approaches as x gets closer and closer to a

Answer

The value of f(a)

Answer

The average value of f(x) over all x in the domain of f

Answer

The derivative of f(x) at x = a

Question 37

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

Accepted Answer

0

Answer

1

Answer

The limit does not exist.

Answer

π

Question 38

Determine whether the function f(x) = ln(x^2 - 4) is continuous on the interval (-∞, -2) U (-2, 2) U (2, ∞).

Accepted Answer

Continuous on (-∞, -2) and (2, ∞) but not on (-2, 2)

Answer

Yes

Answer

No

Answer

Continuous on (-2, 2) but not on (-∞, -2) and (2, ∞)

Question 39

Find the slope of the tangent line to the graph of the function f(x) = x^3 + 2x^2 - 1 at the point (1, 2).

Accepted Answer

7

Answer

1

Answer

3

Answer

5

Question 40

Find the limit of f(x) = sqrt(x^2 + 1) as x goes to infinity.

Accepted Answer

Infinity

Answer

0

Answer

1

Answer

Does not exist

Question 41

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

Accepted Answer

y = 1 and y = -1

Answer

y = 1

Answer

y = -1

Answer

No horizontal asymptotes

Question 42

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

Accepted Answer

x = 2 and x = -2

Answer

x = 2

Answer

x = -2

Answer

No vertical asymptotes

Question 43

Determine whether the function h(x) = |x + 3| is continuous at x = -3. State the theorem or property used to justify your answer.

Accepted Answer

Yes, because the left-hand and right-hand limits exist and are equal.

Answer

Yes, because the graph of the function has no holes or breaks at x = -3.

Answer

No, because the left-hand and right-hand limits do not exist.

Answer

No, because the graph of the function has a sharp corner at x = -3.

Question 44

Find the limit of the function g(x) = (sin x) / x as x approaches infinity. Apply any relevant theorem or technique.

Accepted Answer

0

Answer

1

Answer

undefined

Answer

infinity

Question 45

Which of the following functions is not continuous at x = 0? Identify the type of discontinuity and explain your reasoning.

Accepted Answer

h(x) = 1 / x

Answer

f(x) = x^2

Answer

g(x) = |x|

Answer

k(x) = sin x

Question 46

Find the limit of the function g(x) = (e^x - 1) / x as x approaches 0. Apply L'Hopital's Rule if necessary.

Accepted Answer

1

Answer

0

Answer

e

Answer

infinity

Question 47

State a necessary condition for a function to be differentiable at a point. Explain your reasoning.

Accepted Answer

The function must be continuous at that point.

Answer

The function must have a finite derivative at that point.

Answer

The function's graph must have a sharp corner at that point.

Answer

The function's graph must have a vertical tangent at that point.

Question 48

Determine if the function f(x) = |x - 3| is continuous at x = 3. Explain your reasoning.

Accepted Answer

Discontinuous at x = 3

Answer

Continuous at x = 3

Answer

Continuous from the left at x = 3, but not from the right

Answer

Continuous from the right at x = 3, but not from the left

Question 49

Using the epsilon-delta definition of limits, prove that the limit of the function f(x) = x² - 4 as x approaches 2 is 0.

Accepted Answer

For every ε > 0, there exists a δ > 0 such that |f(x) - 0| < ε whenever 0 < |x - 2| < δ.

Answer

For every ε > 0, there exists a δ > 0 such that |f(x) - 0| > ε whenever 0 < |x - 2| < δ.

Answer

For every ε > 0, there exists a δ > 0 such that |x - 2| < ε whenever 0 < |f(x) - 0| < δ.

Answer

For every ε > 0, there exists a δ > 0 such that |x - 2| > ε whenever 0 < |f(x) - 0| < δ.

Question 50

Determine whether the function f(x) = 1 / (x - 3) is continuous over the entire domain (-∞, ∞). Explain your reasoning.

Accepted Answer

Discontinuous at x = 3

Answer

Continuous on the entire domain

Answer

Discontinuous at x = -3

Answer

Discontinuous at both x = 3 and x = -3

Question 51

Find the limit of the function f(x) = (e^x - 1) / x as x approaches 0 using logarithmic techniques.

Accepted Answer

1

Answer

0

Answer

e

Answer

∞

Question 52

Definition of the Limit of a Function as x approaches a

Accepted Answer

The value that f(x) gets arbitrarily close to as x approaches a

Answer

The value of f(x) when x = a

Answer

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

Answer

The area under the curve of f(x) from x = a to x = b