Limits in Functions: Practice and Test Your Understanding

Question 1

What is the indeterminate form of the following limit: lim as x approaches 0 of (sin(x)) / x?

Accepted Answer

0/0

Answer

∞/∞

Answer

1/0

Answer

0/∞

Question 2

Which of the following statements about the limit of a function as x approaches a specific value c is true?

Accepted Answer

The limit must exist.

Answer

The function must be continuous at c.

Answer

The function must be differentiable at c.

Answer

The function must be defined at c.

Question 3

Use the Squeeze Theorem to evaluate the limit: lim as x approaches 0 of (x² sin(x)) / x

Accepted Answer

0

Answer

1

Answer

∞

Answer

Undefined

Question 4

Find the limit of e^(2x) as x approaches infinity:

Accepted Answer

Infinity

Answer

0

Answer

1

Answer

Does not exist

Question 5

Find the limit of (1 + 1/x)^x as x approaches infinity:

Accepted Answer

e

Answer

0

Answer

1

Answer

Infinity

Question 6

Which of the following theorems is used to evaluate limits involving indeterminate forms of the type 0/0 or infinity/infinity?

Accepted Answer

L'Hopital's rule

Answer

Squeeze theorem

Answer

Intermediate value theorem

Answer

Bolzano-Weierstrass theorem

Question 7

Define a limit.

Accepted Answer

The value a function approaches as its input approaches a certain value

Answer

The value of a function at a certain input

Answer

The rate of change of a function at a certain input

Answer

The integral of a function over a range

Question 8

Is the function f(x) = |x - 3| continuous at x = 3?

Accepted Answer

Discontinuous

Answer

Continuous

Answer

Cannot determine from given information

Question 9

Evaluate the limit: lim (sqrt(x^2 + 1) - x) / (x - 1) as x approaches 1

Accepted Answer

1

Answer

0

Answer

1/2

Answer

Limit does not exist

Question 10

Which condition must be met for a limit to exist at a specific point?

Accepted Answer

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

Answer

The function must be continuous at that point.

Answer

The derivative of the function must exist at that point.

Answer

The function must be bounded in the vicinity of that point.

Question 11

Which theorem can be used to prove that the limit of a sum is equal to the sum of the limits?

Accepted Answer

Algebraic limit theorem

Answer

ε-δ definition of a limit

Answer

Sandwich theorem

Answer

Squeeze theorem

Question 12

Calculate the limit of the function h(x) = sqrt(x² + 4) - x as x approaches infinity.

Accepted Answer

2

Answer

0

Answer

-2

Answer

1

Question 13

Find the horizontal asymptote of the function j(x) = (2x³ + 5x² - 3) / (x³ + 2x² - 1).

Accepted Answer

2

Answer

1

Answer

0

Answer

3

Question 14

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

Accepted Answer

1

Answer

-1

Answer

Undefined

Answer

0

Question 15

Evaluate lim (e^x - 1) / x as x approaches 0.

Accepted Answer

1

Answer

e

Answer

Undefined

Answer

0

Question 16

Calculate the limit of the sequence a_n = (n^2 + 3) / (n^2 - 1) as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

Infinity

Answer

-1

Question 17

What is the limit of the function f(x) = (x^2 - 1)/(x - 1) as x approaches 1?

Accepted Answer

2

Answer

0

Answer

Does not exist

Answer

1

Question 18

Which of the following statements is TRUE about the limit of a function as x approaches a certain value?

Accepted Answer

The limit of a function can exist even if the function is not defined at that value.

Answer

The limit of a function always exists.

Answer

The limit of a function must be equal to the function's value at that point.

Question 19

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

Accepted Answer

1

Answer

∞

Answer

Does not exist

Answer

0

Question 20

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

Accepted Answer

L + M

Answer

L * M

Answer

L - M

Answer

L / M

Question 21

The graph of a function f(x) is shown below. What is the limit of f(x) as x approaches 2?

Accepted Answer

3

Answer

Does not exist

Answer

2

Answer

1

Question 22

What is the limit of the function f(x) = 1/x as x approaches 0 from the right?

Accepted Answer

∞

Answer

Does not exist

Answer

-∞

Answer

0

Question 23

If the limit of f(x) as x approaches a is L, and the limit of g(x) as x approaches a is 0, then what is the limit of f(x)/g(x) as x approaches a?

Accepted Answer

Does not exist

Answer

0

Answer

L/0

Answer

L

Question 24

Which of the following is a key application of limits in calculus?

Accepted Answer

Finding the derivative of a function

Answer

Solving for the roots of a polynomial equation

Answer

Calculating the volume of a solid

Answer

Finding the area under a curve

Question 25

The limit of a function as x approaches infinity represents the function's...

Accepted Answer

Long-term behavior

Answer

Average value

Answer

Minimum value

Answer

Maximum value

Question 26

What is the limit of the function f(x) = (x^3 - 8)/(x - 2) as x approaches 2?

Accepted Answer

12

Answer

0

Answer

Does not exist

Answer

8