Limits and Continuity: A Comprehensive Guide for University Students

Question 1

Clearly and precisely defined: What is the definition of a limit?

Accepted Answer

Balanced and plausible: The value a function approaches as the input gets arbitrarily close to a certain value

Answer

Plausible but less precise: The value a function equals at a specific input

Answer

Plausible but less relevant: The rate of change of a function at a certain input

Answer

Plausible but less relevant: The area under the curve of a function over a certain interval

Question 2

Determine whether the function f(x) = |x| is continuous at x = 0.

Accepted Answer

Corrected: Discontinuous

Answer

Corrected: Continuous

Answer

Simplified: Continuous from the left but not from the right

Answer

Simplified: Continuous from the right but not from the left

Question 3

Given a continuous function f(x) on the interval [0, 1] where f(0) = 0 and f(1) = 1, prove using the Intermediate Value Theorem that there exists a number c between 0 and 1 such that f(c) = 1/2.

Accepted Answer

By the Intermediate Value Theorem, since f(x) is continuous on [0, 1] and f(0) = 0 and f(1) = 1, there exists a number c in (0, 1) such that f(c) = 1/2.

Answer

The Intermediate Value Theorem does not apply here because f(x) is not differentiable on [0, 1].

Answer

Since f(0) = 0 and f(1) = 1, there exists a number c in (0, 1) such that f(c) > 1/2.

Answer

Since f(0) = 0 and f(1) = 1, there exists a number c in (0, 1) such that f(c) < 1/2.

Question 4

Is the function f(x) = {x² if x is rational, x + 1 if x is irrational} continuous on the interval [0, 1]? Choose the best explanation.

Accepted Answer

No, because the function has different values for rational and irrational numbers between 0 and 1.

Answer

Yes, because the function is defined for all values between 0 and 1.

Answer

Yes, because the graph of the function has no sharp corners.

Question 5

According to the $varepsilon$-$delta$ definition of a limit, prove that the limit of $x^2$ as $x$ approaches 2 is equal to 4.

Accepted Answer

If $varepsilon > 0$, then there exists a $delta > 0$ such that if $|x - 2| < delta$, then $|x^2 - 4| < varepsilon$.

Answer

If $varepsilon > 0$, then there exists a $delta > 0$ such that if $|x - 2| < varepsilon$, then $|x^2 - 4| < delta$.

Answer

If $delta > 0$, then there exists an $varepsilon > 0$ such that if $|x - 2| < varepsilon$, then $|x^2 - 4| < delta$.

Answer

If $varepsilon > 0$, then there exists a $delta > 0$ such that if $|x^2 - 4| < varepsilon$, then $|x - 2| < delta$.

Question 6

Identify a function that is discontinuous at a specific point but continuous on the rest of its domain.

Accepted Answer

f(x) = |x - 2|

Answer

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

Answer

f(x) = x^2 - 4

Answer

f(x) = if x > 2 then x + 1 else x - 1

Question 7

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

Accepted Answer

No, because the function has a jump discontinuity at x = 0.

Answer

Yes, because the function is continuous at every point.

Answer

Yes, because the limit of the function as x approaches 0 is 0.

Answer

No, because the function is not differentiable at x = 0.

Question 8

Analyze the function f(x) = (x^3 - 1) / (x - 1). Determine if it has a removable discontinuity, a jump discontinuity, or is continuous at x = 1.

Accepted Answer

Removable discontinuity

Answer

Jump discontinuity

Answer

Continuous at x = 1

Question 9

Which statement about the Intermediate Value Theorem (IVT) is accurate?

Accepted Answer

If f(x) is continuous on [a, b] and c is between f(a) and f(b), then there exists a number d in [a, b] such that f(d) = c.

Answer

If f(x) is continuous on [a, b] and f(a) > f(b), then there exists a number d in [a, b] such that f'(d) = 0.

Answer

If f(x) is continuous on [a, b] and f(a) < f(b), then there exists a number d in [a, b] such that f'(d) > 0.

Answer

If f(x) is differentiable on [a, b] and f'(x) > 0 for all x in (a, b), then f(x) is increasing on [a, b].

Question 10

Define the conditions for continuity of a function f(x) at a point c:

Accepted Answer

f(c) exists and f(c) equals the limit as x approaches c of f(x)

Answer

f(c) exists and the derivative of f(c) is 0

Answer

The derivative of f(c) exists and f(c) equals the limit as x approaches c of f(x)

Answer

The derivative of f(c) exists and the derivative of f(c) is 0

Question 11

For a function to be differentiable at a point, must it also be continuous at that point?

Accepted Answer

Yes, differentiability implies continuity.

Answer

No, differentiability does not imply continuity.

Question 12

Which of the following correctly defines a limit?

Accepted Answer

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

Answer

The value that a function takes on at a specific input value.

Answer

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

Answer

The area under the curve of a function over an interval.

Question 13

Which of the following is NOT an application of the Intermediate Value Theorem?

Accepted Answer

Determining whether a function is increasing or decreasing over an interval.

Answer

Establishing that a function has a root within a specified interval.

Answer

Identifying the maximum and minimum values of a function on an interval.

Answer

Estimating the value of a function using interpolation.

Question 14

What is the general behavior of the limit of a rational function as the denominator approaches infinity?

Accepted Answer

The limit is always 0.

Answer

The limit is always infinity.

Answer

The limit is equal to the numerator.

Answer

The limit is equal to the denominator.

Question 15

Identify the most effective technique for evaluating limits involving trigonometric functions.

Accepted Answer

A combination of these techniques

Answer

Factoring

Answer

Rationalization

Answer

L'Hopital's rule

Question 16

Which of the following statements describes a continuous function?

Accepted Answer

The function has no discontinuities.

Answer

The function is defined at every point in its domain.

Answer

The function is differentiable everywhere.

Answer

The graph of the function is smooth.

Question 17

Prove or disprove using the Intermediate Value Theorem: The function f(x) = x^3 - 2x + 1 has a root between x = 0 and x = 1.

Accepted Answer

Prove

Answer

Disprove

Answer

Cannot be determined

Answer

Not applicable

Question 18

Determine if the function f(x) = x / |x| is continuous at x = 0.

Accepted Answer

No

Answer

Cannot be determined from the given information.

Answer

Yes, because the one-sided limits exist.

Answer

Yes, because the function is differentiable.

Question 19

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

Accepted Answer

f(x) = 1 / x

Answer

f(x) = |x|

Answer

f(x) = e^x

Answer

f(x) = x^2

Question 20

Determine if the function f(x) = { x^2 if x < 2, 3x - 5 if x >= 2 } is continuous at x = 2.

Accepted Answer

No

Answer

Cannot be determined from the given information.

Answer

Yes, because the one-sided limits exist.

Answer

Yes, because the function is piecewise smooth.

Question 21

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

Accepted Answer

Continuous

Answer

Discontinuous jump

Answer

Discontinuous removable

Answer

Discontinuous infinite

Question 22

Calculate the limit of the sequence {a_n} = {n / (n + 1)} as n approaches infinity.

Accepted Answer

1

Answer

Infinity

Answer

Does not exist

Answer

0

Question 23

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

Accepted Answer

f(x) = sin(x)

Answer

f(x) = 1/x

Answer

f(x) = x² - 1

Answer

f(x) = |x| - 1

Question 24

Determine the value of the limit of the function f(x) = (x - 3)/(x² - 9) as x approaches 3.

Accepted Answer

1/6

Answer

0

Answer

Infinity

Answer

Does not exist

Question 25

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

Accepted Answer

Continuity at that point

Answer

Existence of a derivative at that point

Answer

Zero derivative at that point

Answer

Non-zero derivative at that point

Question 26

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

Accepted Answer

1

Answer

0

Answer

Infinity

Answer

Does not exist

Question 27

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

Accepted Answer

e

Answer

0

Answer

∞

Answer

1

Question 28

Apply the Intermediate Value Theorem to determine if the function f(x) = x^3 - 1 has a root between 0 and 1.

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Question 29

Determine if the function f(x) = |x + 2| + |x - 2| is differentiable at x = 0.

Accepted Answer

Not differentiable

Answer

Differentiable at x = 0-

Answer

Differentiable

Answer

Differentiable at x = 0+

Question 30

Which of the following functions is NOT continuous at x = 1?

Accepted Answer

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

Answer

f(x) = e^x

Answer

f(x) = x^2

Answer

f(x) = |x - 2|

Question 31

If f(x) is continuous on [a, b], what does the Intermediate Value Theorem state?

Accepted Answer

For any value y between f(a) and f(b), there exists a c in [a, b] such that f(c) = y.

Answer

The graph of f(x) must intersect the x-axis at least once in [a, b].

Answer

The maximum and minimum values of f(x) must occur at the endpoints of [a, b].

Question 32

Which of the following is NOT a characteristic of a continuous function?

Accepted Answer

It is differentiable at every point in its domain.

Answer

It can be drawn without lifting the pen from the paper.

Answer

Its limit exists as x approaches any value in its domain.

Answer

Its graph has no sudden jumps or breaks.

Question 33

What is the definition of a limit?

Accepted Answer

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

Answer

The derivative of a function.

Answer

The value of a function at a particular input.

Question 34

Use the Intermediate Value Theorem to determine if the function f(x) = x^3 - 1 has a root between 0 and 1.

Accepted Answer

Yes, by the Intermediate Value Theorem

Answer

No

Answer

Cannot be determined

Answer

The function does not have a root.

Question 35

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

Accepted Answer

x = 2

Answer

y = 2

Answer

x = -2

Answer

The function does not have a vertical asymptote.

Question 36

Is the function  *f(x) = x²* uniformly continuous on the interval [0, 1]?

Accepted Answer

Yes

Answer

No, because the function is not bounded on the interval [0, 1].

Answer

No, because the function is not continuous on the interval [0, 1].

Answer

Cannot be determined.

Question 37

Is the function *f(x) = x³ - 2x + 1* differentiable at *x = 1*?

Accepted Answer

Yes

Answer

No, because the function has a sharp corner at *x = 1*.

Answer

Cannot be determined.

Answer

No, because the function is not continuous at *x = 1*.

Question 38

What is the limit of * (x² - 2x + 1) / (x - 1)* as *x* approaches infinity?

Accepted Answer

Infinity

Answer

-1

Answer

2

Answer

1

Question 39

What is true about the continuity of the function *f(x) = x² - 4x + 3*?

Accepted Answer

It is continuous for all real numbers.

Answer

It is discontinuous only at *x = 2*.

Answer

It is continuous only at *x = 2*.

Answer

It is discontinuous at all real numbers.

Question 40

Evaluate the following limit: *lim (x³ - 8) / (x - 2)* as *x* approaches 2.

Accepted Answer

12

Answer

-Infinity

Answer

Infinity

Answer

0

Question 41

Does the function *f(x) = x³ - x + 1* have a root between *x = 0* and *x = 1*? Use the Intermediate Value Theorem.

Accepted Answer

Yes, because *f(0)* is negative and *f(1)* is positive.

Answer

No, because *f(x)* is always decreasing.

Answer

No, because *f(x)* is always increasing.

Question 42

What is the horizontal asymptote of the function *g(x) = (2x² + 3x - 4) / (x - 1)*?

Accepted Answer

y = 2x + 5

Answer

y = 2x - 5

Answer

The function does not have a horizontal asymptote.

Answer

y = 3

Question 43

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

Accepted Answer

f(x) = 1/x

Answer

f(x) = x²

Answer

f(x) = |x|

Question 44

Evaluate the following limit: *lim (sin(3x)) / (2x)* as *x* approaches 0.

Accepted Answer

3/2

Answer

0

Answer

Infinity

Answer

1

Question 45

What are the vertical asymptotes of the function *h(x) = (x - 2) / (x² - 4)*?

Accepted Answer

x = 2 and x = -2

Answer

x = 1 and x = 3

Answer

The function has no vertical asymptotes.

Answer

x = 0 and x = 1

Question 46

Which of the following conditions must be met for a function to be continuous at a point?

Accepted Answer

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

Answer

The function is differentiable at that point.

Answer

The function is increasing at that point.

Answer

The function is bounded at that point.

Question 47

What is the limit as x approaches 0 of the expression (√(x² + 1) - 1) / x?

Accepted Answer

0

Answer

Does not exist

Answer

1/2

Answer

Infinity

Question 48

Evaluate the limit: lim_(x→1) (x² - 1) / (x - 1)

Accepted Answer

2

Answer

0

Answer

Does not exist

Answer

1

Question 49

Find the limit: lim_(x→0) (sin(3x) / x)

Accepted Answer

3

Answer

Does not exist

Answer

1

Answer

0

Question 50

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

Accepted Answer

Does not exist

Answer

1

Answer

-1

Answer

0

Question 51

For what value of k will the function f(x) be continuous at x = 2? 
 f(x) = {x² - 4x + 6,  x < 2; kx - 1, x ≥ 2

Accepted Answer

3

Answer

4

Answer

No such value of k exists

Answer

2

Question 52

Which of the following types of discontinuities is considered removable?

Accepted Answer

Jump discontinuity

Answer

Oscillating discontinuity

Answer

Infinite discontinuity

Answer

All of the above

Question 53

The function f(x) = 1/x is continuous on which of the following intervals?

Accepted Answer

(-∞, 0) and (0, ∞)

Answer

[0, ∞)

Answer

(-∞, ∞)

Question 54

If f(x) is a continuous function on the closed interval [a, b] and f(a) < 0 < f(b), which theorem guarantees that there exists a root of f(x) within the interval (a, b)?

Accepted Answer

Intermediate Value Theorem

Answer

Squeeze Theorem

Answer

Extreme Value Theorem

Answer

Mean Value Theorem

Question 55

What is the limit of (1 + 1/n)^n as n approaches infinity?

Accepted Answer

e

Answer

0

Answer

Infinity

Answer

1

Question 56

The function f(x) = √(x - 4) is continuous for:

Accepted Answer

x ≥ 4

Answer

x > 4

Answer

x ≤ 4

Answer

All real numbers

Question 57

Which of the following statements is NOT true about continuous functions?

Accepted Answer

If f(x) is continuous at x = a, then f(x) is differentiable at x = a.

Answer

If f(x) is continuous at x = a, then the limit of f(x) as x approaches a exists.

Answer

If f(x) and g(x) are continuous at x = a, then f(x) + g(x) is also continuous at x = a.