Introduction to Limits: Interactive Learning Quizzes for Calculus I

Question 1

Let f(x) = |x|, g(x) = -|x|, and h(x) = 0. Use the Squeeze Theorem to find the limit of f(x) as x approaches 0.

Accepted Answer

0

Answer

-1

Answer

1

Answer

The limit does not exist

Question 2

Which of the following accurately defines the limit of a function f(x) as x approaches a?

Accepted Answer

The value that f(x) approaches as x gets indefinitely close to a

Answer

The value that f(a) approaches as x approaches a

Answer

The value that f(x) assumes at x = a

Answer

The value that f(a) assumes at x = 0

Question 3

Identify the option that does NOT represent a limit law:

Accepted Answer

Chain Rule: lim(f(g(x))) = lim(f(g(a)))

Answer

Sum Law: lim(f(x) + g(x)) = lim(f(x)) + lim(g(x))

Answer

Product Law: lim(f(x) * g(x)) = lim(f(x)) * lim(g(x))

Answer

Quotient Law: lim(f(x) / g(x)) = lim(f(x)) / lim(g(x))

Question 4

Determine the horizontal asymptote(s) of the function f(x) = (x^2 + 1) / (x - 1)

Accepted Answer

y = 1

Answer

y = -1

Answer

y = 0

Answer

The function has no horizontal asymptotes

Question 5

Identify the vertical asymptote of the function f(x) = (x^2 - 4) / (x - 2).

Accepted Answer

x = 2

Answer

x = -2

Answer

x = 0

Answer

x = 4

Question 6

Which of the following techniques is employed to determine the limit of a function when it yields an indeterminate form?

Accepted Answer

L'Hopital's Rule

Answer

Implicit Differentiation

Answer

Squeeze Theorem

Answer

Chain Rule

Question 7

Determine whether the function g(x) = 1/(x - 3) has a vertical asymptote at x = 3. Explain your reasoning.

Accepted Answer

True

Answer

The function has a horizontal asymptote at y = 0.

Answer

False

Answer

The function has a removable discontinuity at x = 3.

Question 8

Identify the function that has an infinite discontinuity at x = 0.

Accepted Answer

1 / x^2

Answer

x^3

Answer

sin(x)

Answer

x^2 - 1

Question 9

Which technique is used to evaluate the limit of a function that has an indeterminate form of 0/0?

Accepted Answer

L'Hopital's rule

Answer

Substitution

Answer

Factoring

Question 10

Which of the following is the formal epsilon-delta definition of a limit?

Accepted Answer

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

Answer

The limit of f(x) as x approaches c is L if and only if f(x) gets arbitrarily close to L as x gets arbitrarily close to c.

Answer

The limit of f(x) as x approaches c is L if and only if f(c) = L.

Question 11

Find the limit of the function (sin(x) - x)/(x^2 - x) as x approaches 0.

Accepted Answer

1/2

Answer

Does not exist

Answer

-1/2

Answer

0

Question 12

Which of the following functions has a removable discontinuity at x = 2?

Accepted Answer

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

Answer

f(x) = |x - 2|

Answer

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

Answer

f(x) = x^2

Question 13

Which of the following is a graphical interpretation of a limit?

Accepted Answer

The limit of a function f(x) as x approaches a is the y-coordinate of the point (a, L) on the graph of f(x).

Answer

The limit of a function f(x) as x approaches a is the slope of the tangent line to the graph of f(x) at x = a.

Question 14

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

Accepted Answer

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

Answer

The value that f(x) approaches as x approaches a from both sides.

Answer

The value of f(a).

Question 15

The function f(x) = |x| is not differentiable at x = 0 because the left-hand limit and right-hand limit of the derivative do not exist.

Accepted Answer

It is not differentiable at x = 0

Answer

It has a cusp at x = 0

Answer

It is continuous at x = 0

Answer

It has a vertical tangent at x = 0

Question 16

Which of the following accurately defines a limit?

Accepted Answer

The value that a function approaches as the input gets arbitrarily close to a specific value

Answer

The derivative of a function

Answer

The value of a function at a given input

Question 17

Which of the following limit laws is applicable to finding the limit of a sum of functions?

Accepted Answer

Sum Law of Limits

Answer

Power Law of Limits

Answer

Product Law of Limits

Answer

Quotient Law of Limits

Question 18

Calculate the limit of f(x) = 1 / (x - 2) as x approaches 2.

Accepted Answer

Does not exist

Answer

2

Answer

0

Answer

1

Question 19

What is the formal definition of a limit?

Accepted Answer

The value that a function approaches as the input approaches a given value from either side.

Answer

The average rate of change of a function

Answer

The maximum or minimum value of a function

Question 20

Which of the following limit laws states that the limit of a product is equal to the product of the limits?

Accepted Answer

Product Law

Answer

Sum Law

Answer

Power Law

Answer

Quotient Law

Question 21

What is the limit of a function f(x) as x approaches a specific value?

Accepted Answer

The value that f(x) approaches as x gets arbitrarily close to a, but not necessarily equal to a.

Answer

The slope of the tangent line to the graph of f(x) at x = a.

Answer

The value of f(x) at x = a.

Question 22

Find lim_(x→4) ((x^2 - 16) / (x - 4)).

Accepted Answer

8

Answer

0

Answer

Does not exist

Answer

4

Question 23

Evaluate lim_(x→1) (x^2 - 1) / (x - 1).

Accepted Answer

2

Answer

Infinity

Answer

1

Question 24

What is the necessary condition for a function to have a limit as x approaches a?

Accepted Answer

Both the left-hand limit and the right-hand limit at x = a must exist and be equal.

Answer

The function must be continuous at x = a.

Answer

The function must be defined at x = a.

Question 25

What characterizes a vertical asymptote of a function *f(x)* at *x = a*?

Accepted Answer

As *x* approaches *a*, *f(x)* approaches positive or negative infinity.

Answer

As *x* approaches *a*, *f(x)* approaches a finite value.

Answer

The function is undefined at *x = a*.

Question 26

If the limit of *f(x)* as *x* approaches 3 is 4, but *f(3) = 1*, what does the graph of *f(x)* look like at *x = 3*?

Accepted Answer

The graph has a hole at (3, 1).

Answer

The graph has a vertical asymptote at *x = 3*.

Answer

The function is continuous at *x = 3*.

Answer

The limit does not exist at *x = 3*.

Question 27

When is the Squeeze Theorem useful for determining limits?

Accepted Answer

When a function is bounded by two other functions whose limits are equal at a specific point.

Answer

When the function is differentiable at the point in question.

Answer

When the function is continuous at the point in question.

Question 28

What is the formal definition of the limit of a function *f(x)* as *x* approaches *a*?

Accepted Answer

lim_(x->a) f(x) = L if for any ε > 0, there exists a δ > 0 such that |f(x) - L| < ε whenever 0 < |x - a| < δ

Answer

lim_(x->a) f(x) = L if f(a) = L

Question 29

Which limit law states that lim_(x->a) [f(x) + g(x)] = lim_(x->a) f(x) + lim_(x->a) g(x)?

Accepted Answer

Sum Rule

Answer

Chain Rule

Answer

Quotient Rule

Answer

Product Rule

Question 30

Evaluate the limit: lim_(x->2) (x^2 - 4) / (x - 2)

Accepted Answer

4

Answer

2

Answer

0

Answer

undefined

Question 31

Find the limit of the function as *x* approaches infinity: lim_(x->∞) (2x - 1) / (x + 3)

Accepted Answer

2

Answer

undefined

Answer

0

Answer

1

Question 32

What technique is commonly used to evaluate the limit of a function with an indeterminate form of 0/0?

Accepted Answer

L'Hôpital's Rule

Answer

Direct Substitution

Answer

Conjugate Pairs

Answer

Factoring

Question 33

Determine the horizontal asymptote of the function: f(x) = (x^2 + 1) / (x - 1)

Accepted Answer

y = x + 1

Answer

The function has no horizontal asymptote

Answer

y = x - 1

Answer

y = 1

Question 34

Which of the following is a property of limits?

Accepted Answer

The limit of a constant function is equal to the constant.

Answer

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

Answer

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

Question 35

Find the vertical asymptote of the following function: f(x) = 1 / (x - 2)

Accepted Answer

x = 2

Answer

x = 1

Answer

The function has no vertical asymptote

Answer

x = -1

Question 36

Which of the following limits is an example of the Squeeze Theorem?

Accepted Answer

lim_(x->0) (x^2 * sin(1/x)) = 0

Answer

lim_(x->1) (x^2 - 1) / (x - 1) = 2

Answer

lim_(x->∞) (x^2 + 1) / (x + 1) = ∞

Answer

lim_(x->2) (x - 2) / (x^2 - 4) = 1/2

Question 37

Find the limit of the following function as x approaches 0: lim_(x->0) |x|

Accepted Answer

0

Answer

1

Answer

undefined

Answer

∞

Question 38

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

Accepted Answer

2

Answer

Does not exist

Answer

1

Answer

0

Question 39

The limit of a function as x approaches a point 'a' exists if and only if:

Accepted Answer

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

Answer

The function is continuous at 'a'.

Answer

The function is defined at 'a'.

Question 40

Which of the following limit laws is NOT true?

Accepted Answer

lim (f(x) + g(x)) = lim f(x) + lim g(x) only if both limits exist.

Answer

lim (f(x) / g(x)) = lim f(x) / lim g(x) if both limits exist and lim g(x) ≠ 0.

Answer

lim (f(x) * g(x)) = lim f(x) * lim g(x) if both limits exist.

Question 41

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

Accepted Answer

1

Answer

∞

Answer

0

Answer

Does not exist

Question 42

Given the function f(x) = {x^2 if x ≠ 2, 3 if x = 2}. What is the limit of f(x) as x approaches 2?

Accepted Answer

4

Answer

3

Answer

2

Question 43

Which of the following statements is TRUE regarding the limit of a function?

Accepted Answer

A limit can exist even if the function is not defined at the point in question.

Answer

A limit only exists if the function is continuous at the point in question.

Answer

A limit always exists for any function.

Question 44

The limit of f(x) = 1/x as x approaches 0 is:

Accepted Answer

Does not exist

Answer

1

Answer

∞

Answer

0

Question 45

The limit of f(x) = x^2 + 2x - 3 as x approaches -3 is:

Accepted Answer

0

Answer

Does not exist

Answer

-6

Answer

6

Question 46

If the limit of f(x) as x approaches 'a' is 'L', what can we say about the value of f(a)?

Accepted Answer

f(a) may or may not be equal to 'L'.

Answer

f(a) must be different from 'L'.

Answer

f(a) must be equal to 'L'.

Answer

f(a) must be undefined.

Question 47

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

Accepted Answer

Does not exist

Answer

0

Answer

1

Answer

-1

Question 48

What is the mathematical definition of a limit?

Accepted Answer

The value that a function approaches as the input approaches a specific value

Answer

The value of a function at a specific input

Answer

The slope of a function at a specific input

Question 49

Which of the following is NOT a limit law?

Accepted Answer

The quotient law (a/b = c/d)

Answer

The sum law (lim(a+b) = lim(a) + lim(b))

Answer

The product law (lim(ab) = lim(a) * lim(b))

Answer

The power law (lim(x^n) = lim(x)^n)

Question 50

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

Accepted Answer

4

Answer

0

Answer

6

Answer

Undefined

Question 51

Which of the following techniques can be used to evaluate lim(x^3 - 8) / (x - 2) as x approaches 2?

Accepted Answer

Factoring and canceling

Answer

None of the above

Answer

Squeeze theorem

Answer

L'Hopital's rule

Question 52

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

Accepted Answer

Does not exist

Answer

1

Answer

2

Answer

0

Question 53

Which of the following is true about the limit of f(x) = sin(x) / x as x approaches 0?

Accepted Answer

The limit exists and is equal to 1.

Answer

The limit is equal to infinity.

Answer

The limit does not exist because the function is discontinuous at x = 0.

Question 54

What is the limit of g(x) = (x^2 - 9) / (x - 3) as x approaches 3?

Accepted Answer

6

Answer

Undefined

Answer

9

Answer

0

Question 55

Which of the following is a one-sided limit?

Accepted Answer

lim(x^2) as x approaches 0 from the right

Answer

lim(sin(x)) as x approaches pi/2

Answer

lim(x^2) as x approaches 0

Answer

lim(e^x) as x approaches infinity

Question 56

What is the relationship between the limit of a function at a point and the continuity of the function at that point?

Accepted Answer

If the limit of a function at a point exists and is equal to the value of the function at that point, then the function is continuous at that point.

Answer

If the limit of a function at a point exists, then the function is continuous at that point.

Question 57

Which of the following statements is true about limits and infinite limits?

Accepted Answer

The limit of a function can be positive infinity, negative infinity, or a finite value.

Answer

The limit of a function can only be a finite value.

Answer

The limit of a function can only be positive or negative infinity.