Limits in Functions: Interactive Quizzes and Practice Questions

Question 1

Define the concept of a limit in mathematics.

Accepted Answer

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

Answer

The maximum value a function can attain.

Answer

The rate of change of a function at a specific point.

Answer

The total area under the curve of a function.

Question 2

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

Accepted Answer

1

Answer

0

Answer

e

Answer

Infinity

Question 3

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

Accepted Answer

f(x) = 1 / x

Answer

f(x) = |x|

Answer

f(x) = x^2 - 4

Answer

f(x) = e^x

Question 4

Find the limit of the sequence: a_n = (n + 2) / (2n - 1) as n approaches infinity

Accepted Answer

1

Answer

0

Answer

2

Answer

Infinity

Question 5

Apply the Squeeze Theorem to evaluate the limit: lim(x->0) |x| sin(1/x)

Accepted Answer

0

Answer

1

Answer

Infinity

Answer

The limit does not exist

Question 6

Determine whether the function j(x) = (x^2 - 4) / (x - 2) has a removable discontinuity at x = 2.

Accepted Answer

Yes, the function has a removable discontinuity at x = 2.

Answer

Yes, the function has a non-removable discontinuity at x = 2.

Answer

No, the function is continuous at x = 2.

Answer

No, the function is discontinuous but not removable at x = 2.

Question 7

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

Accepted Answer

0

Answer

1

Answer

Infinity

Answer

The limit does not exist.

Question 8

Find the limit: lim (e^x - 2) / (x - ln 2) as x approaches ln 2

Accepted Answer

1

Answer

0

Answer

2

Answer

Infinity

Question 9

Which of the following statements about the limit of a rational function is accurate?

Accepted Answer

The limit may or may not exist, depending on the specific function.

Answer

The limit is always finite.

Answer

The limit is always infinite.

Answer

The limit is always zero.

Question 10

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

Accepted Answer

2

Answer

4

Answer

0

Answer

1

Question 11

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

Accepted Answer

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

Answer

No, because the left-hand limit does not exist.

Answer

No, because the right-hand limit does not exist.

Answer

Yes, because the function is differentiable at x = 0.

Question 12

Use the epsilon-delta definition to prove that the limit of f(x) = x^2 as x approaches 2 is 4.

Accepted Answer

For any ε > 0, there exists a δ > 0 such that if 0 < |x - 2| < δ, then |f(x) - 4| < ε.

Answer

For any &epsilon; > 0, there exists a &delta; > 0 such that if |x - 2| < &delta;, then |f(x) - 4| > &epsilon;.

Answer

For any δ > 0, there exists an ε > 0 such that if 0 < |x - 2| < δ, then |f(x) - 4| < ε.

Answer

The limit does not exist.

Question 13

Find the limit of the function f(x) = (sin(x) - x) / (x^2 - 1) as x approaches 0.

Accepted Answer

0

Answer

1

Answer

1/2

Answer

The limit does not exist.

Question 14

Find the vertical asymptote of the function f(x) = 1 / (x - 3).

Accepted Answer

x = 3

Answer

x = 0

Answer

x = 2

Answer

The function does not have a vertical asymptote.

Question 15

Determine the type of discontinuity at x = 1 for the function f(x) = {x^2 if x <= 1, 2x - 1 if x > 1}.

Accepted Answer

Jump discontinuity

Answer

Removable discontinuity

Answer

Infinite discontinuity

Answer

Essential discontinuity

Question 16

Who developed the first rigorous definition of a limit using epsilon-delta notation?

Accepted Answer

Cauchy and Weierstrass

Answer

Riemann

Answer

Euler

Answer

Newton

Question 17

Determine the limit of sin(x)/x as x approaches 0.

Accepted Answer

1

Answer

0

Answer

π/2

Answer

Does not exist

Question 18

Evaluate the limit of (e^x - 1)/x as x approaches infinity.

Accepted Answer

1

Answer

0

Answer

e

Answer

∞

Question 19

Find the limit of ln(x^2 + 1) as x approaches infinity.

Accepted Answer

∞

Answer

0

Answer

1

Answer

-∞

Question 20

Precise Definition of a Limit: Which statement best describes the definition of a limit?

Accepted Answer

As the input approaches a specific value or infinity, the function approaches a specific value.

Answer

The exact value of a function at a specific input.

Answer

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

Answer

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

Question 21

Identifying the False Property of Limits: Which property of limits is NOT true?

Accepted Answer

The limit of a quotient is the quotient of the limits.

Answer

The limit of a sum is the sum of the limits.

Answer

The limit of a product is the product of the limits.

Answer

The limit of a constant function is the constant.

Question 22

Continuity Analysis at a Specific Point: Is the function f(x) = |x - 2| continuous at x = 2?

Accepted Answer

Yes, because the limit exists and equals f(2).

Answer

Yes, because the function is differentiable at x = 2.

Answer

No, because the function is not differentiable at x = 2.

Answer

No, because the limit does not exist.

Question 23

Identifying a Vertical Asymptote: Which of the following functions has a vertical asymptote at x = 3?

Accepted Answer

f(x) = 1 / (x - 3)

Answer

f(x) = x^2 - 9

Answer

f(x) = |x - 3|

Answer

f(x) = sin(x - 3)

Question 24

Find the limit of the function f(x) = sqrt(x^2 - 1) as x approaches infinity.

Accepted Answer

Infinity

Answer

0

Answer

1

Answer

-Infinity

Question 25

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 26

Which of the following functions has an infinite discontinuity at x = 2?

Accepted Answer

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

Answer

f(x) = sqrt(x - 2)

Answer

f(x) = |x - 2|

Answer

f(x) = x^2 - 4

Question 27

Which statement accurately describes the definition of a limit?

Accepted Answer

The limit of a function is the value it closely approximates as the input approaches a specific value or approaches infinity.

Answer

The limit of a function is necessarily a finite number.

Answer

The limit of a function can only be evaluated at points where the function is continuous.

Answer

The limit of a function is independent of the path taken to approach the input value.

Question 28

True or False: The limit of the function f(x) = e^x as x approaches infinity is always positive.

Accepted Answer

True

Answer

False

Question 29

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

The limit depends on the value of x.

Question 30

Which of the following sequences converges?

Accepted Answer

a_n = n / (n + 1)

Answer

a_n = 1 + (-1)^n

Answer

a_n = 2^n

Answer

a_n = sin(n)

Question 31

Which condition is essential for the existence of a limit?

Accepted Answer

The left-hand limit and right-hand limit must exist and be equal.

Answer

The function must be continuous at the given point.

Answer

The function must have a continuous derivative at the given point.

Answer

The function must be differentiable at the given point.

Question 32

Identify the indeterminate form in the following limit: lim (x -> 0) (x^2 - 1) / (x - 1).

Accepted Answer

0/0

Answer

∞/∞

Answer

0*∞

Answer

1*∞

Question 33

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

Accepted Answer

1

Answer

0

Answer

2

Answer

Undefined

Question 34

True or False: A function can have a limit at a point even if it is discontinuous at that point.

Accepted Answer

True

Answer

False

Question 35

Which of the following is a graphical representation of the statement 'lim(x->a) f(x) = L'?

Accepted Answer

The graph of f(x) approaches the horizontal line y = L as x gets closer and closer to a.

Answer

The graph of f(x) intersects the horizontal line y = L at x = a.

Answer

The graph of f(x) is always equal to y = L for x greater than a.

Answer

The graph of f(x) is always equal to y = L for x less than a.

Question 36

Evaluate the following limit using L'Hopital's Rule: lim(x->0) (1 - cos(x))/x^2

Accepted Answer

1/2

Answer

0

Answer

1

Answer

Does not exist

Question 37

Find the limit of the function g(x) = (sin x - x) / (x^2 - 1) as x approaches 0. Apply L'Hopital's rule to resolve the indeterminate form.

Accepted Answer

1

Answer

Undefined

Answer

-1

Answer

0

Question 38

Determine the limit of the function h(x) = (x^3 + 2x^2 - 5x + 2) / (x^2 - 1) as x approaches infinity. Use polynomial division or synthetic division to simplify the expression and evaluate the limit.

Accepted Answer

x

Answer

x^2

Answer

0

Answer

Infinity

Question 39

Find the limit of the function h(x) = (1 + 1/x)^x as x approaches infinity. Use the natural logarithm and properties of limits to rewrite the expression and evaluate the limit.

Accepted Answer

e

Answer

Infinity

Answer

0

Answer

1

Question 40

Evaluate the limit of the function f(x) = x * sin(1/x) as x approaches 0. Use the squeeze theorem to determine the limit.

Accepted Answer

0

Answer

Undefined

Answer

1

Answer

-1

Question 41

Find the limit of the function h(x) = sin(x) / x as x approaches infinity.

Accepted Answer

0

Answer

-1

Answer

1

Answer

Infinity

Question 42

Evaluate the limit of the function k(x) = x^2 / (x^2 - 4) as x approaches 2.

Accepted Answer

1

Answer

Infinity

Answer

-1

Answer

0

Question 43

Evaluate the limit of the function g(x) = (sqrt(x + 1) - sqrt(x + 3)) / (x - 2) as x approaches 1.

Accepted Answer

1/8

Answer

Infinity

Answer

-1/8

Answer

0

Question 44

Find the limit of the function f(x) = (x^2 + 3x + 2) / (x + 2) as x approaches -2.

Accepted Answer

7

Answer

-7

Answer

Infinity

Answer

0

Question 45

Evaluate the limit of the function g(x) = (ln(x) - ln(x - 1)) / (x - 1) as x approaches 1.

Accepted Answer

1

Answer

Infinity

Answer

0

Answer

-1

Question 46

What happens to the function f(x) = 1/x as x gets closer and closer to 0?

Accepted Answer

The function approaches positive or negative infinity, depending on which side of 0 x approaches.

Answer

The function approaches 1.

Answer

The function approaches a specific finite value.

Answer

The function approaches 0.

Question 47

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

LM

Answer

L/M

Answer

L - M

Question 48

What is the limit of f(x) = (x^2 - 9) / (x + 3) as x approaches -3?

Accepted Answer

-6

Answer

0

Answer

6

Answer

Does not exist

Question 49

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

Accepted Answer

4

Answer

0

Answer

-4

Answer

2

Question 50

Find the limit of g(x) = (x^3 - 1) / (x - 1) as x approaches 1.

Accepted Answer

3

Answer

0

Answer

1

Answer

-1

Question 51

What is the limit of the function h(x) = sqrt(x^2 + 4) - x as x approaches infinity?

Accepted Answer

2

Answer

0

Answer

-2

Answer

Infinity

Question 52

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

Accepted Answer

1

Answer

e

Answer

0

Answer

Does not exist

Question 53

What is the horizontal asymptote of the function k(x) = (2x^3 + x^2 - 3) / (x^3 - 2)?

Accepted Answer

y = 2

Answer

y = 0

Answer

The function has no horizontal asymptote.

Answer

y = 1

Question 54

Does the function m(x) = (x^2 - 9) / (x - 3) have a vertical asymptote at x = 3?

Accepted Answer

Yes

Answer

Horizontal asymptote at y = 3

Answer

No

Answer

Vertical asymptote at x = -3

Question 55

What is the limit of the function n(x) = (sin(x) - x) / (x - pi/2) as x approaches pi/2?

Accepted Answer

-1

Answer

pi/2

Answer

0

Answer

1

Question 56

Determine the limit of the function p(x) = (x^2 - 25) / (x - 5) as x approaches infinity.

Accepted Answer

Infinity

Answer

-Infinity

Answer

5

Answer

0

Question 57

Find the limit of the function q(x) = (cos(2x) - 1) / (x - pi/4) as x approaches pi/4.

Accepted Answer

2

Answer

1

Answer

0

Answer

-2

Question 58

Determine if the function r(x) = (x^3 - 8) / (x - 2) is continuous at x = 2.

Accepted Answer

Yes

Answer

Continuous at x = 0

Answer

No

Answer

Continuous except at x = 2