Limits of Functions: Interactive Quizzes and Practice Questions

Question 1

Find the limit of the function o(x) = (ln x - 1) / (x - e) as x approaches e.

Accepted Answer

0

Answer

1

Answer

-1

Answer

The limit is undefined.

Question 2

Determine whether the function p(x) = (x^2 - 1) / (x - 1) has a removable discontinuity at x = 1.

Accepted Answer

Yes, it has a removable discontinuity at x = 1.

Answer

No, it does not have a removable discontinuity at x = 1.

Answer

The function has an essential discontinuity at x = 1.

Question 3

Find the limit: lim_(x->3) (x^3 - 27) / (x^2 - 9)

Accepted Answer

0

Answer

3

Answer

9

Answer

27

Question 4

Determine the limit: lim_(x->infinity) (e^x + x^2) / (2e^x - x^3)

Accepted Answer

0

Answer

1/2

Answer

1

Answer

infinity

Question 5

Determine the limit: lim_(x->0) (1 - cos(x)) / x^2

Accepted Answer

1/2

Answer

0

Answer

1

Answer

infinity

Question 6

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

Accepted Answer

e

Answer

1

Answer

0

Question 7

Find the removable discontinuity of the function f(x) = (x² - 1) / (x - 1).

Accepted Answer

x = 1

Answer

x = 0

Answer

x = -1

Answer

The function has no removable discontinuities.

Question 8

Find the horizontal asymptote of the function f(x) = (2x^2 + 3x - 5) / (x^2 - 1). Horizontal asymptotes are found by evaluating limits at infinity.

Accepted Answer

y = 2

Answer

y = -2

Answer

y = 3

Answer

y = 5

Question 9

The limit of a function f(x) as x approaches a is the value L such that:

Accepted Answer

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

Answer

f(x) is continuous at x = a.

Answer

f(a) exists.

Answer

f(x) is differentiable at x = a.

Question 10

Which of the following is NOT a valid limit evaluation technique?

Accepted Answer

Plugging in the value of x directly.

Answer

L'Hopital's rule

Answer

Factoring and cancellation

Answer

Rationalization

Question 11

Evaluate the limit: lim (x^2 - 4) / (x - 2) as x approaches 2.

Accepted Answer

4

Answer

Does not exist

Answer

0

Answer

2

Question 12

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

Accepted Answer

lim (f(x) / g(x)) as x approaches a = lim (f'(x) / g'(x)) as x approaches a, provided that the limit of f(x) / g(x) is indeterminate of type 0/0 or ∞/∞.

Answer

lim (f(x) / g(x)) as x approaches a = lim (f'(x) - g'(x)) as x approaches a

Answer

lim (f(x) / g(x)) as x approaches a = lim (f'(x) + g'(x)) as x approaches a

Question 13

Find the limit: lim (e^x - 1) / x as x approaches 0.

Accepted Answer

1

Answer

0

Answer

e

Question 14

What is the horizontal asymptote of the function f(x) = (x^2 + 1) / (x - 1)?

Accepted Answer

y = 1

Answer

The function does not have a horizontal asymptote

Answer

y = 0

Answer

x = 1

Question 15

Which of the following functions has a vertical asymptote at x = 2?

Accepted Answer

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

Answer

f(x) = e^x

Answer

f(x) = x^2 + 2x

Answer

f(x) = sin(x)

Question 16

Evaluate the limit: lim (x → 2) (x² - 4) / (x - 2)

Accepted Answer

0

Answer

Undefined

Answer

4

Answer

2

Question 17

Find the limit: lim (x -> 2) f(x), where f(x) = { x² if x < 2, x+1 if x ≥ 2 }

Accepted Answer

5

Answer

4

Answer

3

Question 18

Find the limit: lim (x → 1) (x + sqrt(x)) / (x - 1)

Accepted Answer

3/2

Answer

2

Answer

∞

Answer

1

Question 19

Evaluate the limit: lim x->0 (x^2 - 4x + 3) / (x - 1)

Accepted Answer

1

Answer

3

Answer

0

Answer

2

Question 20

Find the limit: lim x->2 (sin(x - 2) / (x - 2))

Accepted Answer

1

Answer

0

Answer

cos(2)

Answer

π

Question 21

Find the limit: lim x->0 (e^x - 1) / x

Accepted Answer

1

Answer

0

Answer

e

Answer

∞

Question 22

Evaluate the limit: lim x->-∞ (x^3 - 2x^2 + 1) / (x^2 - 4)

Accepted Answer

-∞

Answer

1

Answer

0

Answer

∞

Question 23

Find the limit: lim x->π/2 (cos(x) / (x - π/2))

Accepted Answer

0

Answer

π/2

Answer

∞

Answer

1

Question 24

Determine the limit: lim x->1 (x - 1) / (x^2 - 1)

Accepted Answer

1/2

Answer

1

Answer

∞

Answer

0

Question 25

Evaluate the limit: lim n->∞ (1 + 2/n)^(n^2)

Accepted Answer

e^4

Answer

∞

Answer

1

Answer

2