Limits at Infinity: Explore Asymptotes and Infinite Limits

Question 1

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

Accepted Answer

1

Answer

0

Answer

Infinity

Answer

Does not exist

Question 2

Determine the limit of the function h(x) = (x² - 1)/(x³ - 1) as x approaches positive infinity.

Accepted Answer

1

Answer

0

Answer

Infinity

Answer

Does not exist

Question 3

Find the horizontal asymptote of the function f(x) = (3x^2 - 2x + 1) / (x^2 + 1).

Accepted Answer

y = 1

Answer

y = 0

Answer

y = 2

Answer

y = 3

Question 4

Find the limit of the function h(x) = (5x^3 + 2x^2 - 1) / (3x^3 - x^2 + 2) as x approaches infinity.

Accepted Answer

5/3

Answer

3/5

Answer

2/1

Answer

1/0

Question 5

Identify the horizontal asymptote of the function f(x) = (e^x - 1) / (e^x + 1).

Accepted Answer

y = 1

Answer

y = 0

Answer

y = e

Answer

y = -1

Question 6

Find the horizontal asymptote of the function f(x) = (3x^2 - 2x + 1) / (x^2 - 1).

Accepted Answer

y = 3

Answer

y = 2

Answer

y = 0

Answer

y = 1

Question 7

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

Accepted Answer

e

Answer

0

Answer

Infinity

Answer

1

Question 8

Determine the vertical asymptote of the function n(x) = (x^2 - 9) / (x - 3), for all real numbers x except x = 3.

Accepted Answer

x = 3

Answer

x = 0

Answer

x = 9

Answer

x = -3

Question 9

Determine the vertical asymptotes of the function q(x) = (x^2 - 4) / (x^2 + 2x), for all real numbers x except x = 0 and x = -2.

Accepted Answer

x = 0, x = -2

Answer

x = 2

Answer

x = 0

Answer

x = -4

Question 10

Determine the horizontal asymptote of the function s(x) = (e^x + x) / (e^x - x), as x approaches infinity.

Accepted Answer

y = 1

Answer

y = 0

Answer

y = -1

Question 11

Find the horizontal asymptote of the function f(x) = (3x^2 + 4x - 5) / (2x^2 - 7x + 3).

Accepted Answer

y = 3/2

Answer

y = -3/2

Answer

y = 1/3

Question 12

Determine the vertical asymptote(s) of the function g(x) = (x^2 - 4) / (x - 2).

Accepted Answer

x = 2

Answer

y = 2

Answer

x = -2

Answer

No vertical asymptotes

Question 13

Which of the following functions has a horizontal asymptote at y = 2 and a vertical asymptote at x = 3?

Accepted Answer

f(x) = (2x + 1) / (x - 3)

Answer

f(x) = (x - 2) / (x + 3)

Answer

f(x) = (x + 2) / (x - 3)

Answer

f(x) = (x^2 + 1) / (x - 3)

Question 14

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

Accepted Answer

1

Answer

0

Answer

-1

Question 15

Find the limit of the function g(x) = (x^3 + 2x^2 - 1) / (x^4 - 16) as x approaches infinity.

Accepted Answer

0

Answer

∞

Answer

-∞

Answer

1

Question 16

What is the limit as x approaches infinity of the function f(x) = (3x^2 + 2x) / (x^2 - 1)?

Accepted Answer

3

Answer

0

Answer

∞

Answer

-1

Question 17

The function f(x) = 1/x has a vertical asymptote at:

Accepted Answer

x = 0

Answer

x = 1

Answer

y = 1

Answer

y = 0

Question 18

Which of the following functions has a horizontal asymptote at y = 0?

Accepted Answer

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

Answer

f(x) = 1/x

Answer

f(x) = x^2 + 1

Answer

f(x) = x

Question 19

What is the limit of the function $f(x) = \frac{3x^2 - 5x + 2}{x^2 + 1}$ as $x$ approaches infinity?

Accepted Answer

3

Answer

$\infty$

Answer

-$\infty$

Answer

0

Question 20

Which of the following functions has a horizontal asymptote at $y = 2$?

Accepted Answer

$f(x) = \frac{2x^2 + 1}{x^2 - 3}$

Answer

$f(x) = \frac{x}{x + 2}$

Answer

$f(x) = \frac{x^3 + 1}{x^2}$

Question 21

Determine the limit: $\lim_{x \to \infty} \frac{\sqrt{4x^2 + 3}}{2x - 1}$

Accepted Answer

1

Answer

0

Answer

$\infty$

Answer

2

Question 22

The function $f(x) = \frac{1}{x - 2}$ has a vertical asymptote at:

Accepted Answer

$x = 2$

Answer

$x = 0$

Answer

$y = 0$

Answer

There is no vertical asymptote.

Question 23

A population of bacteria grows according to the model $P(t) = \frac{1000}{1 + 9e^{-0.5t}}$ where $t$ is time in hours. What is the long-term behavior of the population?

Accepted Answer

The population approaches 1000.

Answer

The population grows exponentially without bound.

Answer

The population approaches 0.

Answer

The population oscillates between 0 and 1000.

Question 24

Find the horizontal asymptote(s) of the function $f(x) = \frac{\sin(x)}{x}$.

Accepted Answer

$y = 0$

Answer

$x = 0$

Answer

There are no horizontal asymptotes.

Question 25

Determine $\lim_{x \to -\infty} (3 - \frac{2}{x} + \frac{4}{x^2})$.

Accepted Answer

3

Answer

-$\infty$

Answer

$\infty$

Answer

0

Question 26

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

Accepted Answer

Infinity

Answer

0

Answer

-4

Answer

2

Question 27

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

Accepted Answer

1/2

Answer

Infinity

Answer

-1/2

Answer

0

Question 28

Which of the following is the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches negative infinity?

Accepted Answer

-Infinity

Answer

0

Answer

4

Answer

Infinity

Question 29

Find the vertical asymptote(s) of the function g(x) = (x^2 - 9) / (x^2 - 4).

Accepted Answer

x = 2 and x = -2

Answer

x = 0 and x = 3

Answer

There are no vertical asymptotes.

Answer

x = 3 and x = -3

Question 30

Which of the following functions has a vertical asymptote at x = 1 and a horizontal asymptote at y = 2?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 31

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

Accepted Answer

1/2

Answer

-1/2

Answer

0

Question 32

What is the limit of the function f(x) = (3x^2 + 2x - 1) / (x^2 - 4) as x approaches infinity?

Accepted Answer

3

Answer

0

Answer

∞

Question 33

Which of the following functions has a horizontal asymptote at y = 2?

Accepted Answer

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

Answer

f(x) = (x^3 + 2x) / (x^2 - 1)

Answer

f(x) = (x^2 + 1) / (x^3 - 1)

Answer

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

Question 34

The function f(x) = 1 / (x - 2) has a vertical asymptote at x = ?

Accepted Answer

2

Answer

-2

Answer

1

Answer

0

Question 35

What is the limit of the function f(x) = e^(-x) as x approaches infinity?

Accepted Answer

0

Answer

1

Answer

-∞

Answer

∞

Question 36

A company's revenue, R(x), is modeled by the function R(x) = 1000x / (x + 10), where x represents the number of units sold. What is the company's maximum possible revenue as x approaches infinity?

Accepted Answer

1000

Answer

500

Answer

∞

Answer

0

Question 37

Given the function f(x) = (x^2 - 1) / (x + 1), which of the following statements is true about the function's behavior as x approaches -1?

Accepted Answer

The function has a vertical asymptote at x = -1.

Answer

The function approaches infinity.

Answer

The function approaches 0.

Answer

The function has a hole at x = -1.

Question 38

Which of the following functions has no horizontal asymptotes?

Accepted Answer

f(x) = x^3 + 2x

Answer

f(x) = (x^2 + 1) / (x^3 - 1)

Answer

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

Answer

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

Question 39

The limit of the function f(x) = sin(x) / x as x approaches infinity is ?

Accepted Answer

0

Answer

1

Answer

Does not exist

Answer

∞

Question 40

A company's profit, P(x), is modeled by the function P(x) = 500x - x^2, where x represents the number of units sold. As the company produces and sells more and more units (x approaches infinity), what happens to the profit?

Accepted Answer

Profit approaches negative infinity.

Answer

Profit approaches a constant value.

Answer

Profit approaches positive infinity.

Answer

Profit remains unchanged.

Question 41

The function f(x) = (x - 1) / (x^2 - 1) has a hole at x = ?

Accepted Answer

1

Answer

2

Answer

0

Answer

-1

Question 42

What is the limit of the function f(x) = 3x^2 / (x^2 + 1) as x approaches infinity?

Accepted Answer

3

Answer

-Infinity

Answer

Infinity

Answer

0

Question 43

Which of the following functions has a horizontal asymptote at y = 2?

Accepted Answer

f(x) = (2x^2 + 3) / (x^2 - 1)

Answer

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

Answer

f(x) = e^(-x) + 2

Answer

f(x) = x^3 / (x^2 + 1)

Question 44

Determine the limit of g(x) = (4x - 1) / (x + 2) as x approaches negative infinity.

Accepted Answer

4

Answer

0

Answer

-2

Answer

Infinity

Question 45

The function h(x) = 1 / (x - 3) has a vertical asymptote at:

Accepted Answer

x = 3

Answer

y = 0

Answer

x = 0

Answer

y = 1

Question 46

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

Accepted Answer

4

Answer

Infinity

Answer

Does Not Exist

Answer

2

Question 47

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

Accepted Answer

f(x) approaches negative infinity

Answer

f(x) approaches 0

Answer

f(x) approaches -2

Answer

f(x) approaches positive infinity

Question 48

A population of bacteria grows according to the model P(t) = 1000 / (1 + 9e^(-0.5t)), where t is time in hours. What is the limiting population size as time approaches infinity?

Accepted Answer

1000

Answer

100

Answer

0

Answer

Infinity

Question 49

The function f(x) = (sin x) / x has a limit as x approaches infinity. What is the value of this limit?

Accepted Answer

0

Answer

1

Answer

Does Not Exist

Answer

Infinity

Question 50

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

Accepted Answer

L + M

Answer

L / M

Answer

L - M

Answer

L * M

Question 51

The horizontal asymptote of a function f(x) can be found by analyzing the:

Accepted Answer

Degrees of the highest power terms in the numerator and denominator

Answer

y-intercept of the function

Answer

x-intercepts of the function