Limits in Functions: Master the Concept with Interactive Quizzes

Question 1

Determine the limit of the function h(x) = (3x^2 + 2x - 1) / (x^2 - 1) as x approaches infinity.

Accepted Answer

3

Answer

1

Answer

0

Answer

Does not exist

Question 2

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

Accepted Answer

1

Answer

0

Answer

1/4

Answer

Does not exist

Question 3

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

Accepted Answer

No, it does not exist.

Answer

Yes, it exists and is equal to 0.

Answer

Yes, it exists and is equal to 1.

Answer

The existence of the limit depends on the value of x.

Question 4

evaluate the limit of the function sin(x) / x as x approaches 0, if it exists

Accepted Answer

0

Answer

-1

Answer

The limit does not exist

Question 5

Determine the limit of the function g(x) = (sin(x) - x) / (x³ - x) as x approaches 0.

Accepted Answer

0

Answer

1

Answer

-1

Answer

Does not exist

Question 6

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

Accepted Answer

1

Answer

0

Answer

e

Answer

Infinity

Question 7

Determine whether the function f(x) = 1/(x - 2) has a limit as x approaches 2.

Accepted Answer

No, the limit does not exist.

Answer

Yes, the limit is 0.

Answer

Yes, the limit is 1.

Answer

Yes, the limit is infinity.

Question 8

Find the limit of the function g(x) = (sqrt(x² + 1) - x) / (x - 1) as x approaches 1.

Accepted Answer

1/2

Answer

0

Answer

1

Answer

Does not exist

Question 9

Determine the limit of the function h(x) = (sin(2x)) / x as x approaches 0.

Accepted Answer

0

Answer

1

Answer

2

Answer

Does not exist

Question 10

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

Accepted Answer

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

Answer

The value of f(a) as x approaches a

Answer

The value of f(x) as x approaches infinity

Answer

The value of f(x) as x approaches zero

Question 11

Which of the following is a necessary condition for the existence of a limit?

Accepted Answer

The function must be defined at the point.

Answer

The function must be continuous at the point.

Answer

The function must have a finite value at the point.

Answer

The function must be differentiable at the point.

Question 12

Find the limit of the function f(x) = (x^3 - 8) / (x^2 - 4) as x approaches 2.

Accepted Answer

2

Answer

4

Answer

6

Answer

The limit does not exist

Question 13

Determine if the function f(x) is continuous at x = 3, given that f(x) = x^2 if x < 3, 5 if x = 3, and 3x - 6 if x > 3

Accepted Answer

No

Answer

Yes

Answer

Cannot be determined

Question 14

Find the limit of the function (x² + 2x - 3) / (x + 3) as x approaches -3 using synthetic division. Show your work.

Accepted Answer

-2

Answer

-1

Answer

0

Answer

1

Question 15

What is the most precise definition of a limit?

Accepted Answer

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

Answer

The exact value of the function at a specific input value.

Answer

The highest or lowest value that the function can reach.

Question 16

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

Accepted Answer

x = 3

Answer

x = 2

Answer

x = 4

Answer

The function does not have a vertical asymptote.

Question 17

Which of the following correctly describes the behavior of the function f(x) = 1 / x^2 as x approaches infinity?

Accepted Answer

It approaches 0.

Answer

It approaches infinity.

Answer

It approaches 1.

Answer

It has no limit.

Question 18

Which of the following is a necessary condition for the existence of a limit?

Accepted Answer

The left-hand and right-hand limits must be equal.

Answer

The derivative of the function must be continuous at the point.

Answer

The function must be defined at the point.

Answer

The function must be continuous at the point.

Question 19

Find the limit of the function f(x) = |x - 3| as x approaches 3.

Accepted Answer

0

Answer

1

Answer

2

Answer

3

Question 20

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

Accepted Answer

1

Answer

0

Answer

e

Answer

Infinity

Question 21

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

Accepted Answer

2

Answer

0

Answer

1

Answer

The limit does not exist

Question 22

Which of the following is the correct form of L'Hopital's rule?

Accepted Answer

lim (f(x) / g(x)) = lim (f'(x) / g'(x))

Answer

lim (f(x) - g(x)) = lim (f'(x) - g'(x))

Answer

lim (f(x) * g(x)) = lim (f'(x) * g'(x))

Answer

lim (f(x)^g(x)) = lim (f'(x)^g'(x))

Question 23

Which of the following functions has a removable discontinuity at x = 2?

Accepted Answer

f(x) = (x - 2) / (x^2 - 4)

Answer

g(x) = (x^2 - 4) / (x - 2)

Answer

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

Answer

k(x) = x^2 - 4

Question 24

Find the limit of the function f(x) = |x - 2| as x approaches 2 from the right.

Accepted Answer

2

Answer

0

Answer

-2

Answer

Does not exist

Question 25

Determine the vertical asymptote of the function k(x) = (x - 3) / (x^2 - 9).

Accepted Answer

x = 3

Answer

x = -3

Answer

x = 0

Answer

The function has no vertical asymptote.

Question 26

Which of the following is a sufficient condition for the existence of a limit at x = c?

Accepted Answer

The left-hand limit and the right-hand limit are equal.

Answer

The function is continuous at x = c.

Answer

The function is differentiable at x = c.

Answer

The function is defined at x = c.

Question 27

Which of the following statements about limits is true?

Accepted Answer

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

Answer

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

Answer

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

Answer

The limit of a composition of functions is always the composition of the limits.

Question 28

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

Accepted Answer

0

Answer

2

Answer

4

Answer

The limit does not exist.

Question 29

Find the vertical asymptote of the function g(x) = 1 / (x - 5).

Accepted Answer

x = 5

Answer

x = 0

Answer

x = -5

Answer

The function does not have a vertical asymptote.

Question 30

What is the definition of a limit in mathematics?

Accepted Answer

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

Answer

The precise value a function reaches at a particular input

Answer

The rate of change of a function at a specific input

Answer

The region beneath a curve within a specified interval

Question 31

Which theorem is employed to calculate the limit of a function resulting in the indeterminate form 0/0?

Accepted Answer

L'Hopital's Rule

Answer

Squeeze Theorem

Answer

Intermediate Value Theorem

Answer

Extreme Value Theorem

Question 32

Describe the behavior of the graph of a function with a limit at infinity

Accepted Answer

Approaches a horizontal line as x approaches infinity

Answer

Approaches a vertical line as x approaches infinity

Answer

Approaches a parabolic curve as x approaches infinity

Answer

Approaches a circle as x approaches infinity

Question 33

Evaluate the limit: lim(x->0) tan(x)

Accepted Answer

0

Answer

1

Answer

∞

Answer

The limit does not exist

Question 34

Define a limit in the context of functions.

Accepted Answer

When the value of the function gets arbitrarily close to a certain value as the input approaches a specific value.

Answer

When the output of the function equals a specific value at a specific input value.

Answer

When the derivative of the function approaches a specific value as the input approaches a specific value.

Answer

When the area under the curve of the function over a specific interval approaches a specific value.

Question 35

Find the limit of (sin(x) - x) / (x^3 - x) as x approaches 0. Use L'Hopital's rule to evaluate this limit.

Accepted Answer

1/6

Answer

-1/6

Answer

0

Answer

Undefined

Question 36

Determine the limit of (1 + 1/x)^x as x approaches infinity. Use a logarithmic transformation to evaluate this limit.

Accepted Answer

e

Answer

Undefined

Answer

0

Answer

1

Question 37

Find the limit of (x + sqrt(x^2 + 1)) / (x + 1) as x approaches infinity. Factor and simplify the expression to evaluate the limit.

Accepted Answer

1

Answer

∞

Answer

0

Answer

Undefined

Question 38

What is the limit of the function  *f(x) = (x³ - 8) / (x - 2)* as *x* approaches 2?

Accepted Answer

12

Answer

0

Answer

8

Answer

Does Not Exist

Question 39

Which of the following describes a function with a vertical asymptote at *x = 3*?

Accepted Answer

The limit of the function as *x* approaches 3 is either positive or negative infinity.

Answer

The limit of the function as *x* approaches 3 is 0.

Answer

The function is continuous at *x = 3*.

Answer

The function is undefined at *x = 3*.

Question 40

Evaluate the limit of the function *f(x) = (x² - 1) / (x - 1)* as *x* approaches 1.

Accepted Answer

2

Answer

1

Answer

0

Answer

Does Not Exist

Question 41

If the limit of *f(x)* as *x* approaches *c* is *L*, and the limit of *g(x)* as *x* approaches *c* is *M*, what is the limit of *f(x) + g(x)* as *x* approaches *c*?

Accepted Answer

L + M

Answer

L - M

Answer

L * M

Answer

L / M

Question 42

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

Accepted Answer

1

Answer

0

Answer

e

Answer

Does Not Exist

Question 43

Find the limit of the function *f(x) = (x² - 1) / (x - 1)* as *x* approaches 1.

Accepted Answer

2

Answer

Undefined

Answer

0

Answer

1

Question 44

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

Accepted Answer

1

Answer

0

Answer

Undefined

Answer

e

Question 45

Which of the following is true about the limit of a function as the input approaches infinity?

Accepted Answer

It may exist, be infinite, or not exist

Answer

It must always be finite

Answer

It must always be zero

Answer

It must always exist

Question 46

Which of the following techniques can be used to evaluate the limit of a function that is indeterminate at a particular value?

Accepted Answer

Factoring, rationalization, or L'Hopital's rule

Answer

Estimation

Answer

Substitution

Answer

Graphing

Question 47

Which of the following is NOT a property of limits?

Accepted Answer

Limits are always differentiable

Answer

Limits are unique

Answer

Limits can be used to analyze the behavior of functions

Answer

Limits can be used to find asymptotes

Question 48

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

Accepted Answer

0

Answer

1

Answer

-1

Answer

Undefined

Question 49

Which of the following statements is true about the limit of a piecewise function?

Accepted Answer

It may exist if the limits from both sides are equal

Answer

It never exists

Answer

It always exists