Limits and Continuity: Master Calculus Concepts with Interactive Quizzes

Question 1

What is the formal definition of the limit of a function $f(x)$ as $x$ approaches $a$?

Accepted Answer

The value to which $f(x)$ gets arbitrarily close as $x$ approaches $a$, or $lim_{x o a} f(x) = L$, where for any $epsilon > 0$, there exists a $delta > 0$ such that $|f(x) - L| < epsilon$ whenever $0 < |x - a| < delta$.

Answer

The value of $f(x)$ at $x = a$, or $lim_{x 	o a} f(x) = f(a)$.

Answer

The value of $x$ such that $f(x) = a$, or $lim_{x 	o a} f(x) = x$ such that $f(x) = a$.

Answer

The value of $f(a)$, or $lim_{x 	o a} f(x) = f(a)$.

Question 2

Evaluate the limit: lim(x→2) (x^2 - 4) / (x - 2)

Accepted Answer

0

Answer

2

Answer

4

Answer

6

Question 3

Find the slope of the tangent line to the curve y = 2x^2 + 3x - 5 at the point (1, 0).

Accepted Answer

7

Answer

1

Answer

5

Answer

9

Question 4

Which of the following functions is continuous at every real number?

Accepted Answer

f(x) = |x|

Answer

f(x) = 1 / x

Answer

f(x) = tan(x)

Answer

f(x) = sqrt(x)

Question 5

Find the limit of the following sequence: {n^2 + 3n} / (n^3 + 2n)

Accepted Answer

1

Answer

0

Answer

1/2

Answer

1/3

Question 6

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

Accepted Answer

3x^2

Answer

3

Answer

x^2

Answer

1

Question 7

Analyze the motion of a particle given by the function s(t) = t^3 - 3t^2 + 2t. Find the intervals where the particle is moving in the positive direction.

Accepted Answer

(0, 1) and (2, ∞)

Answer

(-∞, 0) and (2, ∞)

Answer

(0, 2)

Answer

(0, 2) and (1, ∞)

Question 8

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

Accepted Answer

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

Answer

The function must be differentiable at that point

Answer

The function must have a finite value at that point

Answer

The graph of the function must be smooth at that point

Question 9

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

Accepted Answer

Discontinuous

Answer

Continuous

Answer

Cannot be determined from the provided information

Question 10

Which theorem is used to prove the continuity of a function on an interval?

Accepted Answer

Intermediate Value Theorem

Answer

Extreme Value Theorem

Answer

Squeeze Theorem

Answer

Chain Rule

Question 11

Which of the following is NOT a type of discontinuity?

Accepted Answer

Continuous discontinuity

Answer

Removable discontinuity

Answer

Jump discontinuity

Answer

Infinite discontinuity

Question 12

Which function is NOT differentiable at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = x^3

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 13

Which theorem can be used to prove that the function f(x) = x^3 is continuous at any point a?

Accepted Answer

Polynomial Continuity Theorem

Answer

Intermediate Value Theorem

Answer

Chain Rule

Answer

Squeeze Theorem

Question 14

Which condition is NOT necessary for a function f(x) to be continuous at x = a?

Accepted Answer

f(a) is rational

Answer

lim(x→a) f(x) exists

Answer

f(a) exists

Answer

lim(x→a) f(x) = f(a)

Question 15

Which limit represents the instantaneous rate of change of the function f(x) at x = 2?

Accepted Answer

lim(h→0) [f(2 + h) - f(2)] / h

Answer

lim(x→2) [f(x) - f(2)] / (x - 2)

Answer

lim(h→0) [f(2) - f(2 + h)] / h

Answer

lim(x→2) [f(2 + h) - f(2)] / (x - 2)

Question 16

The limit of a function f(x) as x approaches a is the value that f(x) **approaches** as x **gets closer and closer** to a.

Accepted Answer

The value that f(x) takes at x = a

Answer

The derivative of f(x) at x = a

Answer

The integral of f(x) over the interval [a, b]

Answer

The maximum value of f(x) on the interval [a, b]

Question 17

Which of the following functions is **discontinuous** at x = 1?

Accepted Answer

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

Answer

f(x) = x^2

Answer

f(x) = |x - 2|

Answer

f(x) = sin(x)

Question 18

If f(x) is continuous on an interval, then the graph of f(x) will have no **breaks** or **jumps** on that interval.

Accepted Answer

True

Answer

False

Answer

Sometimes true, sometimes false

Question 19

If f(x) is continuous at x = a and g(x) is continuous at x = f(a), then the **composition** function g(f(x)) is **continuous** at x = a.

Accepted Answer

True

Answer

False

Answer

Sometimes true, sometimes false

Question 20

The **Intermediate Value Theorem** states that if a function is continuous on an interval, then it takes on all values between its **minimum** and **maximum** values on that interval.

Accepted Answer

True

Answer

False

Answer

Sometimes true, sometimes false

Question 21

A function f(x) is **differentiable** at a point a if the **limit** of the **difference quotient** (f(x) - f(a)) / (x - a) as x **approaches** a exists.

Accepted Answer

True

Answer

False

Answer

Sometimes true, sometimes false

Question 22

The **integral** of the function f(x) = x^3 from 0 to 1 is **1/4**.

Accepted Answer

True

Answer

False

Answer

Sometimes true, sometimes false

Question 23

Which statement is a necessary condition for a function to be differentiable at a point?

Accepted Answer

The function must be continuous at that point.

Answer

The function must be continuous on an interval containing that point.

Answer

The function must have a derivative at that point.

Answer

The function must have a second derivative at that point.

Question 24

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

Accepted Answer

f(x) = 1/x

Answer

f(x) = sin(x)

Answer

f(x) = x^2

Answer

f(x) = e^x

Question 25

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

Accepted Answer

Continuous

Answer

Discontinuous

Answer

Cannot be determined without more information.

Question 26

Determine the continuity of the function f(x) = { x^2 if x ≤ 1, 2x - 1 if x > 1 } at x = 1.

Accepted Answer

Discontinuous

Answer

Continuous

Answer

Cannot be determined without more information.

Question 27

Find the derivative of the function f(x) = ln(x^2 + 1).

Accepted Answer

2x/(x^2 + 1)

Answer

x/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

(x^2 + 1)/x

Question 28

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

Accepted Answer

1

Answer

0

Answer

e

Answer

The limit does not exist.

Question 29

Determine the continuity of the function f(x) = sqrt(x - 1) at x = 1.

Accepted Answer

Discontinuous

Answer

Continuous

Answer

Cannot be determined without more information.

Question 30

For a function to have a removable discontinuity at a point, which condition must be met?

Accepted Answer

The limit of the function at the point exists.

Answer

The function is continuous at the point.

Answer

The function has a vertical asymptote at the point.

Answer

The function has a jump discontinuity at the point.

Question 31

Define the limit of a function.

Accepted Answer

The value the function approaches as the input gets arbitrarily close to a specific value

Answer

The value the function equals at a specific input value

Answer

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

Answer

The maximum or minimum value of the function

Question 32

Identify the function that is discontinuous at x = 0.

Accepted Answer

f(x) = 1/x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = |x|

Question 33

Describe how L'Hopital's rule is used to evaluate the limit of a function as the input approaches a specific value.

Accepted Answer

Differentiate the numerator and denominator of the function and then evaluate the limit of the resulting expression

Answer

Integrate the numerator and denominator of the function and then evaluate the limit of the resulting expression

Answer

Take the reciprocal of the function and then evaluate the limit of the resulting expression

Answer

Use the squeeze theorem to bound the function between two other functions

Question 34

Which of the following is **not** a necessary condition for a function to be continuous at a point?

Accepted Answer

The function has a vertical asymptote at the point.

Answer

The function is defined at the point.

Answer

The limit of the function as x approaches the point exists.

Answer

The left- and right-hand limits of the function at the point are **equal**.

Question 35

Find the point of discontinuity of the function f(x) = (x**2 - 9) / (x - 3).

Accepted Answer

x = 3

Answer

x = 0

Answer

x = -3

Answer

The function is continuous everywhere.

Question 36

Find the vertical asymptotes of the function f(x) = (x**2 - 4) / (x - 2).

Accepted Answer

x = 2

Answer

x = -2

Answer

x = 0

Answer

The function has no vertical asymptotes.

Question 37

Find the points of discontinuity of the function f(x) = { x**2 if x ≤ 0, x + 1 if x > 0 }.

Accepted Answer

x = 0

Answer

x = 1

Answer

The function is continuous everywhere.

Answer

x = -1

Question 38

Which of the following accurately defines the limit of a function f(x) as x approaches a?

Accepted Answer

The value L such that given any fixed positive number ε > 0, there exists a positive number δ > 0 such that whenever 0 < |x - a| < δ, then |f(x) - L| < ε

Answer

The value of f(a)

Question 39

Find the limit of the function (x^2 - 1) / (x - 1) as x approaches 1

Accepted Answer

2

Answer

0

Answer

1

Question 40

Determine whether the function f(x) = |x - 5| is continuous at x = 5

Accepted Answer

True

Answer

False

Answer

Cannot be determined

Question 41

Find the horizontal asymptote of the function f(x) = (2x + 3) / (x - 2)

Accepted Answer

y = 2

Answer

y = -2

Answer

y = x + 2

Answer

y = x

Question 42

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

Accepted Answer

x = ±1

Answer

x = 0

Answer

x = -2

Answer

x = 2

Question 43

Which of the following theorems guarantees that if a function is continuous on a closed interval, then it attains both its minimum and maximum values on that interval?

Accepted Answer

Extreme Value Theorem

Answer

Intermediate Value Theorem

Answer

Rolle's Theorem

Question 44

Find the limit of the sequence {a_n}, where a_n = (2n^3 + 5n^2 - 3) / (n^3 + 2n^2 + 1) as n approaches infinity

Accepted Answer

2

Answer

∞

Answer

1

Answer

0

Question 45

Determine whether the series ∑(n=1 to ∞) (n / 2^n) is convergent or divergent

Accepted Answer

Convergent

Answer

Divergent

Answer

None of the above

Answer

Cannot be determined

Question 46

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

Accepted Answer

2

Answer

1

Answer

0

Answer

Limit does not exist

Question 47

For what value of the constant 'a' is the function f(x) = {x^3 - a, if x < 1; 2x + 1, if x ≥ 1} continuous at x = 1?

Accepted Answer

1

Answer

2

Answer

3

Answer

0

Question 48

Use L'Hôpital's rule to find the limit of the function f(x) = (sin x)/x as x approaches 0.

Accepted Answer

1

Answer

∞

Answer

Limit does not exist

Answer

0

Question 49

Determine whether the function f(x) = x^2 - 4 is uniformly continuous on the interval [-2, 2].

Accepted Answer

True

Answer

Depends on the value of ε

Answer

Cannot determine

Answer

False

Question 50

Find the limit of the function f(x) = e^(x^2) as x approaches infinity.

Accepted Answer

∞

Answer

Limit does not exist

Answer

0

Answer

1

Question 51

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

Accepted Answer

1/2

Answer

0

Answer

Limit does not exist

Answer

∞