Master Limit Properties: Enhance Your Calculus Skills

Question 1

Which statement is NOT a limit law?

Accepted Answer

The limit of a function is equal to the value of the function at that point, if the function is continuous at that point.

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 quotient is the quotient of the limits, provided the denominator does not approach zero.

Question 2

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 3

Which of the following is a sufficient condition for a limit to exist?

Accepted Answer

The function must be continuous at the point.

Answer

The function must be defined at the point.

Answer

The function must have a finite value at the point.

Answer

The function must be differentiable at the point.

Question 4

Which of the following is an example of a limit that does not exist?

Accepted Answer

lim (x^2) / x as x approaches 0

Answer

lim (sin(x)) / x as x approaches 0

Answer

lim (e^x) / x as x approaches infinity

Answer

lim (x) / (x^2 - 1) as x approaches infinity

Question 5

Which statement is NOT a property of limits?

Accepted Answer

The limit of a function always exists.

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 quotient is the quotient of the limits.

Question 6

Which of the following is **not** a property of limits?

Accepted Answer

Limits are defined for all functions

Answer

The limit of a constant times a function is the constant times the limit of the function

Answer

The limit of a function divided by another function is the limit of the first function divided by the limit of the second function

Answer

The limit of a sum is the sum of the limits

Question 7

Find the limit: lim (x->0) (sin x) / x

Accepted Answer

1

Answer

0

Answer

x

Answer

Does not exist

Question 8

Use the squeeze theorem to evaluate the limit: lim (x → 0) (x^(2) sin(1/x))

Accepted Answer

0

Answer

-1

Answer

1

Answer

The limit does not exist.

Question 9

Calculate the limit of the expression below as x approaches 0:

lim (sin(x)) / x

Accepted Answer

1

Answer

0

Answer

undefined

Answer

infinity

Question 10

Apply the sum rule of limits to find the limit of the following function as x approaches -1:

lim (3x^2 - 2x + 1)

Accepted Answer

4

Answer

0

Answer

does not exist

Answer

infinity

Question 11

Use the product rule of limits to find the limit of the following expression as x approaches 0:

lim (x^3 * sin(2x))

Accepted Answer

0

Answer

infinity

Answer

does not exist

Answer

1

Question 12

Calculate the limit of the following function as x approaches 1 using L'Hôpital's rule:

lim ((e^(2x) - 1) / (x^2 - 1))

Accepted Answer

2

Answer

1

Answer

infinity

Answer

0

Question 13

Evaluate lim [(sin(3x) / x)] as x approaches 0 using the squeeze theorem. Which of the following functions can be used for squeezing?

Accepted Answer

0 ≤ |sin(3x)| ≤ |3x| ≤ 3|x|

Answer

|3x| ≤ |sin(3x)| ≤ 0 ≤ 3|x|

Answer

0 ≤ 3|x| ≤ |sin(3x)| ≤ |3x|

Answer

0 ≤ |sin(3x)| ≤ 3|x| ≤ |3x|

Question 14

Evaluate lim [(e^x - 1) / (x - 1)] as x approaches 1 using the natural logarithm.

Accepted Answer

e

Answer

Undefined

Answer

1

Answer

0

Question 15

Evaluate lim [(1 - cos(x)) / x^2] as x approaches 0 using l'Hôpital's rule.

Accepted Answer

1/2

Answer

0

Answer

1

Question 16

Find the limit: lim [((1+x)^n - 1) / x] as x approaches 0. Apply the power rule of limits.

Accepted Answer

n

Answer

Undefined

Answer

1

Answer

0

Question 17

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

Undefined

Answer

0

Answer

L - M

Question 18

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 the limit of f(x) + g(x) as x approaches a is:

Accepted Answer

L + M

Answer

LM

Answer

-LM

Answer

L - M

Question 19

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 the limit of f(x)g(x) as x approaches a is:

Accepted Answer

LM

Answer

L/M

Answer

-LM

Answer

M/L

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 (where M is not zero), then the limit of f(x)/g(x) as x approaches a is:

Accepted Answer

L/M

Answer

L - M

Answer

-LM

Answer

M/L

Question 21

The squeeze theorem states that if f(x) ≤ g(x) ≤ h(x) for all x in an open interval containing a, and if the limit of f(x) and the limit of h(x) as x approaches a are both L, then:

Accepted Answer

The limit of g(x) as x approaches a is also L.

Answer

The limit of g(x) as x approaches a is equal to 0.

Answer

The limit of g(x) as x approaches a does not exist.

Question 22

The limit of f(x) as x approaches a is 0. Which of the following statements about f(x) is true?

Accepted Answer

There exists a δ > 0 such that |f(x)| < ε whenever 0 < |x - a| < δ.

Answer

There exists an ε > 0 such that |f(x)| < δ whenever 0 < |x - a| < ε.

Answer

There exists an M > 0 such that |f(x)| > M whenever 0 < |x - a| < δ.

Question 23

Determine whether the sequence a_n = (n² + 1) / (n + 2) converges or diverges.

Accepted Answer

Converges

Answer

Diverges

Answer

Oscillates

Answer

Cannot be determined

Question 24

Identify the statement that is NOT a valid property of limits.

Accepted Answer

Limits are always finite.

Answer

Limits can be simplified using algebraic operations.

Answer

Limits can be evaluated from both the left and the right.

Answer

Limits can be used to determine the convergence of sequences.

Question 25

Which of the following is NOT a property of limits?

Accepted Answer

Addition law does not hold for limits as x approaches infinity.

Answer

Limit of a sum is the sum of limits.

Answer

Limit of a product is the product of limits (when both limits exist).

Question 26

Find the value of the following limit: lim(x→1) √((x² - 1)/(x - 1))

Accepted Answer

√2

Answer

0

Answer

1

Answer

-√2

Question 27

Apply the limit laws to evaluate the limit: lim(x→∞) (x³ - 2x² + 3x - 4)/(x² - x + 1)

Accepted Answer

∞

Answer

-∞

Answer

1

Answer

0

Question 28

Which condition is sufficient to ensure the existence of a limit for a function?

Accepted Answer

The left-hand limit and right-hand limit both exist and are equal.

Answer

The function is continuous at the point of interest.

Answer

The function is differentiable at the point of interest.

Question 29

Use the squeeze theorem to find the limit of the function f(x) = x²sin(1/x) as x approaches 0.

Accepted Answer

0

Answer

-1

Answer

This limit does not exist.

Answer

1

Question 30

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

Accepted Answer

f(x) = x / (x² - 1)

Answer

f(x) = e^x

Answer

f(x) = |x|

Answer

f(x) = 1 / x

Question 31

Find the limit of the function f(x) = (sinx) / x as x approaches 0.

Accepted Answer

1

Answer

This limit does not exist.

Answer

0

Answer

Infinity

Question 32

Evaluate the following limit using the limit laws:
lim (x^2 - 4) / (x - 2)

Accepted Answer

2

Answer

4

Answer

0

Answer

undefined

Question 33

Determine the limit of the following function as x approaches infinity:
lim (sqrt(x^2 + 1) - x)

Accepted Answer

0

Answer

undefined

Answer

1

Answer

-1

Question 34

Find the limit of the following function as x approaches 0 from the right:
lim (|x| - x) / x

Accepted Answer

-1

Answer

undefined

Answer

0

Answer

1

Question 35

Use the squeeze theorem to evaluate the limit:
lim (x^2 - 1) / (x^2 - 2) as x approaches 1

Accepted Answer

1

Answer

2

Answer

undefined

Answer

0

Question 36

Evaluate the following limit using L'Hopital's rule:
lim (e^x - 1) / x as x approaches 0

Accepted Answer

1

Answer

undefined

Answer

0

Answer

e

Question 37

Determine whether the following limit exists:
lim (sin(x) / x) as x approaches infinity

Accepted Answer

1

Answer

0

Answer

undefined

Answer

-1

Question 38

Evaluate the following limit using the limit laws for functions:
lim ((sqrt(x + 2) - 2) / (x - 2)) as x approaches 2

Accepted Answer

1 / 4

Answer

0

Answer

undefined

Answer

1 / 2

Question 39

Find the limit of the following function as x approaches 1 from the left:
lim (x^3 - 1) / (x - 1) as x approaches 1-

Accepted Answer

3

Answer

0

Answer

undefined

Answer

1

Question 40

Use the epsilon-delta definition of a limit to prove the following:
lim (x^2 - 1) / (x - 1) = 2 as x approaches 1

Accepted Answer

N/A (proof is not provided in the question)

Answer

N/A

Answer

N/A

Answer

N/A

Question 41

Which of the following properties of limits is NOT true?

Accepted 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.

Answer

If the limits of two functions exist and are equal, then the limits of their compositions are equal.

Question 42

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then what is $\lim_{x \to a} [f(x) + g(x)]$?

Accepted Answer

$L + M$

Answer

$L / M$

Answer

$L - M$

Answer

$L \cdot M$

Question 43

What is $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$?

Accepted Answer

4

Answer

2

Answer

0

Answer

Does not exist

Question 44

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = 0$ where $g(x) \neq 0$ for $x \neq a$, then what is $\lim_{x \to a} \frac{f(x)}{g(x)}$?

Accepted Answer

The limit does not exist.

Answer

$0$

Answer

$\infty$

Answer

$L$

Question 45

If $f(x) \leq g(x) \leq h(x)$ for all $x$ in an open interval containing $a$, except possibly at $x = a$, and $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} h(x) = L$, then what can we conclude about $\lim_{x \to a} g(x)$?

Accepted Answer

$\lim_{x \to a} g(x) = L$

Answer

$\lim_{x \to a} g(x) = \infty$

Answer

We cannot conclude anything about $\lim_{x \to a} g(x)$.

Answer

$\lim_{x \to a} g(x) = 0$

Question 46

What is $\lim_{x \to 0} \frac{sin(x)}{x}$?

Accepted Answer

1

Answer

Does not exist

Answer

0

Answer

$\infty$

Question 47

If $\lim_{x \to a} f(x) = L$ and $c$ is a constant, then what is $\lim_{x \to a} [c \cdot f(x)]$?

Accepted Answer

$c \cdot L$

Answer

$c / L$

Answer

$c + L$

Question 48

What is $\lim_{x \to 0} \frac{1}{x}$?

Accepted Answer

The limit does not exist.

Answer

0

Answer

1

Answer

$\infty$

Question 49

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then what is $\lim_{x \to a} [f(x) - g(x)]$?

Accepted Answer

$L - M$

Answer

$L / M$

Answer

$L + M$

Answer

$L \cdot M$

Question 50

If $f(x)$ is a continuous function at $x = a$, what is $\lim_{x \to a} f(x)$?

Accepted Answer

$f(a)$

Answer

The limit does not exist.

Answer

0

Answer

$\infty$

Question 51

What is $\lim_{x \to \infty} \frac{1}{x}$?

Accepted Answer

0

Answer

1

Answer

The limit does not exist.

Answer

$\infty$