Master Limit Properties: Enhance Your Calculus Skills

Question 1

Determine the limit of a constant function.

Accepted Answer

Equals the constant value

Answer

Is undefined

Answer

Equals infinity

Answer

Equals negative infinity

Question 2

Identify the property of limits that states the limit of a sum equals the sum of the limits.

Accepted Answer

Sum Law

Answer

Product Law

Answer

Quotient Law

Answer

Constant Multiple Law

Question 3

Find the limit of the sequence {n / (n + 1)} as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

Infinity

Answer

Negative infinity

Question 4

Identify the limit property that states the limit of a product equals the product of the limits.

Accepted Answer

Product Law

Answer

Sum Law

Answer

Quotient Law

Answer

Constant Multiple Law

Question 5

Determine the limit of (e^x - 1) / x as x approaches infinity.

Accepted Answer

1

Answer

0

Answer

Infinity

Answer

Negative infinity

Question 6

Determine whether the following limit exists:
$$lim_{x 	o 1} rac{x^2 - 1}{x - 1}$$

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Question 7

Which of the following is the correct definition of a limit?

Accepted Answer

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

Answer

The value of a function at a specific input.

Answer

The rate of change of a function at a specific input.

Answer

The integral of a function over a specific interval.

Question 8

Which of the following is the correct form of the squeeze theorem?

Accepted Answer

If (f(x)) ≤ (g(x)) ≤ (h(x)) and (lim_{x 	o a}g(x) = L) then (lim_{x 	o a}f(x) = lim_{x 	o a}h(x) = L).

Answer

If (f(x)) ≥ (g(x)) ≥ (h(x)) and (lim_{x 	o a}g(x) = L) then (lim_{x 	o a}f(x) = lim_{x 	o a}h(x) = L).

Answer

If (f(x)) ≤ (h(x)) ≤ (g(x)) and (lim_{x 	o a}g(x) = L) then (lim_{x 	o a}f(x) = lim_{x 	o a}h(x) = L).

Answer

If (f(x)) ≥ (h(x)) ≥ g(x) and (lim_{x 	o a}g(x) = L) then (lim_{x 	o a}f(x) = lim_{x 	o a}h(x) = L).

Question 9

Find the limit of the following function as (x) approaches infinity:
$$f(x) = e^x - 2$$

Accepted Answer

Infinity

Answer

0

Answer

1

Answer

-Infinity

Question 10

Which of the following is a property of limits? Select the most comprehensive option.

Accepted Answer

All of the above properties hold true for limits

Answer

The limit of a sum is equal to the sum of the limits

Answer

The limit of a product is equal to the product of the limits

Answer

The limit of a quotient is equal to the quotient of the limits

Question 11

Find the limit: lim ((√(x^2 + 1) - x)/x) as x approaches infinity.

Accepted Answer

0

Answer

1

Answer

1/2

Answer

The limit does not exist

Question 12

Which of the following statements about limits is incorrect?

Accepted Answer

The limit of a composite function is always equal to the product of the limits of the inner and outer functions

Answer

The limit of a constant function is the constant itself

Answer

The limit of a polynomial function is the value of the polynomial at the limit point

Answer

The limit of a rational function is the quotient of the limits of the numerator and denominator

Question 13

Determine whether the limit of the function f(x) = |x - 2| as x approaches 2 exists.

Accepted Answer

Yes

Answer

Limit is 0

Answer

No

Answer

Limit is infinity

Question 14

Determine whether the function f(x) = (x^3 - 1) / (x^2 - 1) has an infinite discontinuity at x = 1.

Accepted Answer

Yes

Answer

No

Answer

Removable discontinuity

Answer

Jump discontinuity

Question 15

Evaluate the limit of the function g(x) = ln(|x|) as x approaches -1.

Accepted Answer

Does not exist

Answer

0

Answer

1

Answer

Infinity

Question 16

Given that 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 equal to 0), what is the limit of the quotient of f(x) divided by g(x) as x approaches 'a'?

Accepted Answer

$\frac{L}{M}$

Answer

$\frac{M}{L}$

Answer

L - M

Answer

M - L

Question 17

Given that the limit of f(x) as x approaches 'a' is L and 'k' is a constant, what is the limit of the product of k and f(x) as x approaches 'a'?

Accepted Answer

kL

Answer

\frac{L}{k}

Answer

k - L

Answer

L - k

Question 18

Given that the limit of f(x) as x approaches 'a' is L and g(x) is defined as the product of a constant 'c' and f(x), what is the limit of g(x) as x approaches 'a'?

Accepted Answer

cL

Answer

\frac{L}{c}

Answer

L - c

Answer

c - L

Question 19

Given that the limit of f(x) as x approaches 'a' is a positive value 'L', and the limit of g(x) as x approaches 'a' is a negative value 'M', what is the limit of the product of f(x) and g(x) as x approaches 'a'?

Accepted Answer

Negative

Answer

Zero

Answer

Positive

Question 20

Given that the limit of f(x) as x approaches 'a' is L and the limit of g(x) as x approaches 'a' is positive infinity, what is the limit of the product of f(x) and g(x) as x approaches 'a'?

Accepted Answer

Infinite

Answer

Finite

Answer

Zero

Answer

Indeterminate

Question 21

Evaluate the limit as x approaches 0 of (x^2 + 2x) / (x - 1).

Accepted Answer

2

Answer

-1

Answer

0

Question 22

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

Accepted Answer

e

Answer

Infinity

Answer

0

Answer

1

Question 23

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

Accepted Answer

It may not exist.

Answer

It must be positive.

Answer

It must exist.

Answer

It must be finite.

Question 24

If f(x) is continuous at x = a, what can be concluded about lim (x -> a) f(x)?

Accepted Answer

It is equal to f(a).

Answer

It is equal to 0.

Answer

It does not exist.

Question 25

If lim (x → a) f(x) = L and lim (x → a) g(x) = M, then lim (x → a) [f(x) + g(x)] equals:

Accepted Answer

L + M

Answer

M - L

Answer

|L - M|

Answer

L - M

Question 26

If lim (x → a) f(x) = ∞ and lim (x → a) g(x) = ∞, then lim (x → a) [f(x) - g(x)] is:

Accepted Answer

Indeterminate

Answer

a

Answer

0

Answer

∞

Question 27

Consider the function f(x) = |x|. What is lim (x → 0) f(x)?

Accepted Answer

0

Answer

-1

Answer

1

Answer

Does not exist

Question 28

Evaluate the limit: lim (x → ∞) (x³ - 2x² + 5) / (x² - 1)

Accepted Answer

∞

Answer

0

Answer

-∞

Answer

5

Question 29

What is the value of lim (x → 0) (sin x) / x?

Accepted Answer

1

Answer

0

Answer

∞

Question 30

Given that the limit of  *f(x)* as *x* approaches *a* is *L* and the limit of *g(x)* as *x* approaches *a* is *M*, what is the limit of  *(f(x) + g(x))* as *x* approaches *a*?

Accepted Answer

L + M

Answer

Cannot be determined

Answer

LM

Answer

L - M

Question 31

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

Accepted Answer

L - M

Answer

ML

Answer

L + M

Answer

Cannot be determined

Question 32

Given that the limit of *f(x)* as *x* approaches *a* is *L* and the limit of *g(x)* as *x* approaches *a* is *M*, what is the limit of *(f(x) * g(x))* as *x* approaches *a*?

Accepted Answer

LM

Answer

Cannot be determined

Answer

L/M

Answer

L + M

Question 33

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

Accepted Answer

L/M

Answer

Cannot be determined

Answer

L - M

Answer

M/L

Question 34

What is the limit of *(1/x)* as *x* approaches infinity?

Accepted Answer

0

Answer

Infinity

Answer

1

Answer

Does not exist

Question 35

Find the limit of  *(x^2 - 1) / (x - 1)* as *x* approaches 1.

Accepted Answer

2

Answer

1

Answer

Does not exist

Answer

0

Question 36

Given that the limit of *f(x)* as *x* approaches *a* is *L*, what is the limit of *f(x + h)* as *x* approaches *a*?

Accepted Answer

L

Answer

Cannot be determined

Answer

L + h

Answer

L - h

Question 37

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

Accepted Answer

0

Answer

Does not exist

Answer

Infinity

Answer

1

Question 38

Given that the limit of *f(x)* as *x* approaches *a* is infinity and the limit of *g(x)* as *x* approaches *a* is infinity, what is the limit of *(f(x) + g(x))* as *x* approaches *a*?

Accepted Answer

Infinity

Answer

0

Answer

Does not exist

Answer

1

Question 39

If  lim_(x->a) f(x) = L and lim_(x->a) g(x) = M, what is the value of lim_(x->a) [f(x) + g(x)]?

Accepted Answer

L + M

Answer

L - M

Answer

L / M

Answer

L * M

Question 40

Evaluate lim_(x->0) sin(x)/x.

Accepted Answer

1

Answer

0

Answer

Does Not Exist

Answer

∞

Question 41

If lim_(x->a) f(x) = L and c is a constant, what is the value of lim_(x->a) [c * f(x)]?

Accepted Answer

c * L

Answer

c - L

Answer

L / c

Answer

c + L

Question 42

Evaluate lim_(x->∞) 1/x.

Accepted Answer

0

Answer

∞

Answer

Does Not Exist

Answer

1

Question 43

If lim_(x->a) f(x) = L and n is a positive integer, then what is the value of lim_(x->a) [f(x)]^n?

Accepted Answer

L^n

Answer

n * L

Answer

L - n

Answer

L + n

Question 44

Given that ( lim_{x 	o a} f(x) = L ) and ( lim_{x 	o a} g(x) = M ), which of the following is equal to ( lim_{x 	o a} [f(x) + g(x)] )?

Accepted Answer

L + M

Answer

LM

Answer

L - M

Question 45

Evaluate ( lim_{x 	o 2} rac{x^2 - 4}{x - 2} ).

Accepted Answer

4

Answer

0

Answer

2

Question 46

If ( lim_{x 	o a} f(x) = L ) and ( lim_{x 	o a} g(x) = M ), then what is the value of ( lim_{x 	o a} [f(x) * g(x)] )?

Accepted Answer

LM

Answer

L + M

Answer

L/M

Answer

L - M

Question 47

If ( lim_{x 	o a} f(x) = L ) and ( lim_{x 	o a} g(x) = M ), where M ≠ 0, then what is the value of ( lim_{x 	o a} rac{f(x)}{g(x)} )?

Accepted Answer

L/M

Answer

L + M

Answer

LM

Answer

L - M

Question 48

Calculate ( lim_{x 	o 0} rac{sin(x)}{x} ).

Accepted Answer

1

Answer

Does not exist

Answer

0

Answer

∞

Question 49

If ( lim_{x 	o a} f(x) = L ) and c is a constant, then what is the value of ( lim_{x 	o a} [c * f(x)] )?

Accepted Answer

cL

Answer

L/c

Answer

c - L

Answer

c + L

Question 50

Find ( lim_{x 	o ∞} rac{1}{x} ).

Accepted Answer

0

Answer

Does not exist

Answer

∞

Answer

1

Question 51

If ( lim_{x 	o a} f(x) = L ) and n is a positive integer, then what is the value of ( lim_{x 	o a} [f(x)]^n )?

Accepted Answer

L^n

Answer

L/n

Answer

nL

Question 52

Determine the value of ( lim_{x 	o 3} rac{x^2 - 9}{x - 3} ) by factoring and simplifying.

Accepted Answer

6

Answer

0

Answer

Does not exist

Answer

3

Question 53

Find ( lim_{x 	o 0} rac{sin(3x)}{2x} ).

Accepted Answer

3/2

Answer

Does not exist

Answer

1

Answer

0