Limits of Functions: Interactive Quizzes and Practice Questions

Question 1

Define the precise mathematical definition of a limit.

Accepted Answer

If the values of f(x) can get arbitrarily close to L, as x approaches a (but not necessarily equal to a), then the limit of f(x) as x approaches a is L.

Answer

If f(x) is defined at x = a, then the limit of f(x) as x approaches a is f(a).

Answer

If f(x) is continuous at x = a, then the limit of f(x) as x approaches a is f(a).

Answer

If the graph of f(x) has a vertical asymptote at x = a, then the limit of f(x) as x approaches a does not exist.

Question 2

Which of the following functions is not differentiable at x = 0?

Accepted Answer

f(x) = |x|

Answer

g(x) = x^2

Answer

h(x) = sin(x)

Answer

j(x) = e^x

Question 3

Find the derivative of the function f(x) = ln(x^2 + 1).

Accepted Answer

2x/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

x/(x^2 + 1)

Answer

2/(x^2 + 1)

Question 4

Consider the function f(x) defined as follows: f(x) = x^2 if x is less than or equal to 1, and f(x) = 2x - 1 if x is greater than 1. Determine the limit of f(x) as x approaches 1.

Accepted Answer

1

Answer

0

Answer

2

Question 5

Define the limit of a function.

Accepted Answer

The value that a function approaches as the input approaches a particular point

Answer

The exact value of a function at a given input

Answer

The rate of change of a function at a particular input

Answer

The minimum or maximum value of a function

Question 6

Which limit theorem states that the limit of a sum is equal to the sum of the limits?

Accepted Answer

Sum Rule

Answer

Product Rule

Answer

Quotient Rule

Answer

Power Rule

Question 7

Which condition is NOT necessary for the existence of a limit?

Accepted Answer

The function must have a discontinuity at the point

Answer

The function must be defined at the point

Answer

The left-hand and right-hand limits must exist and be equal

Answer

The function must be continuous at the point

Question 8

Identify the function that has a removable discontinuity at x = 0:

Accepted Answer

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

Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Question 9

State the correct application of the squeeze theorem:

Accepted Answer

If f(x) ≤ g(x) ≤ h(x) and limx→a f(x) = limx→a h(x) = L, then limx→a g(x) = L

Answer

If f(x) ≥ g(x) ≥ h(x) and limx→a f(x) = limx→a h(x) = L, then limx→a g(x) = L

Answer

If f(x) ≤ g(x) ≤ h(x) and limx→a f(x) = limx→a h(x) = ∞, then limx→a g(x) = ∞

Answer

If f(x) ≥ g(x) ≥ h(x) and limx→a f(x) = limx→a h(x) = -∞, then limx→a g(x) = -∞

Question 10

Interpret the following limit: limx→∞ f(x) = L

Accepted Answer

As x approaches infinity, f(x) approaches L

Answer

As x approaches negative infinity, f(x) approaches L

Answer

When x is very large, f(x) is exactly equal to L

Answer

When x is very small, f(x) is exactly equal to L

Question 11

Which function has an infinite limit at x = 1?

Accepted Answer

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

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 12

State the correct form of the epsilon-delta definition of a limit:

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

Question 13

Let g(x) = (x^2 - 16)/(x - 4). Use the limit definition to find lim_(x->4) g(x).

Accepted Answer

0

Answer

4

Answer

8

Answer

16

Question 14

Provide the formal definition of a limit of a function.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 15

Which limit theorem is applicable for evaluating the limit of a quotient of functions?

Accepted Answer

If lim x->c f(x) = L1 and lim x->c g(x) = L2 ≠ 0, then lim x->c (f(x) / g(x)) = L1 / L2

Answer

If lim x->c f(x) = L1 and lim x->c g(x) = L2, then lim x->c (f(x) + g(x)) = L1 + L2

Answer

If lim x->c f(x) = L1 and lim x->c g(x) = L2, then lim x->c (f(x) - g(x)) = L1 - L2

Answer

If lim x->c f(x) = L1 and lim x->c g(x) = L2, then lim x->c (f(x) * g(x)) = L1 * L2

Question 16

Identify the function that has a discontinuity at x = 3.

Accepted Answer

f(x) = |x - 3|

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 17

Which property of limits states that the limit of a difference of functions is equal to the difference of their limits?

Accepted Answer

If lim x->c f(x) = L and lim x->c g(x) = M, then lim x->c (f(x) - g(x)) = L - M

Answer

If lim x->c f(x) = L and lim x->c g(x) = M, then lim x->c (f(x) + g(x)) = L + M

Answer

If lim x->c f(x) = L and lim x->c g(x) = M ≠ 0, then lim x->c (f(x) / g(x)) = L / M

Answer

If lim x->c f(x) = L and lim x->c g(x) = M, then lim x->c (f(g(x))) = f(g(c))

Question 18

Provide an example of the squeeze theorem.

Accepted Answer

If lim x->c f(x) ≤ g(x) ≤ h(x) and lim x->c f(x) = lim x->c h(x) = L, then lim x->c g(x) = L

Answer

If lim x->c f(x) = lim x->c g(x) = L, then lim x->c (f(x) + g(x)) = 2L

Answer

If lim x->c f(x) = L and lim x->c g(x) = M, then lim x->c (f(x) * g(x)) = L * M

Answer

If lim x->c f(x) = L, then lim x->c (f(x))^2 = L^2

Question 19

Evaluate lim x->∞ ((x^2 + 3x - 2) / (2x^2 - x + 3)).

Accepted Answer

1

Answer

1/2

Answer

3/2

Answer

Does not exist

Question 20

Which condition is insufficient to guarantee the existence of a limit?

Accepted Answer

The function is differentiable at the point

Answer

The function is continuous at the point

Answer

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

Answer

The function is bounded

Question 21

The epsilon-delta definition of a limit is crucial in real analysis because it:

Accepted Answer

Offers a rigorous framework for defining and proving limit theorems.

Answer

Provides a simple way to visualize limits.

Answer

Simplifies the calculation of limits.

Question 22

**Explain the relationship between the existence of a limit of a function at a point and the continuity of the function at that point.**

Accepted Answer

A function is continuous at a point if and only if its limit exists at that point.

Answer

A function is continuous at a point if and only if its limit exists at that point and is infinite.

Answer

A function is continuous at a point if and only if its limit does not exist at that point.

Answer

The continuity of a function at a point is unrelated to the existence of a limit at that point.

Question 23

Identify the correct property of limits of functions:

Accepted Answer

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 L + M.

Accepted Answer

If f(x) is continuous at x = a, then the limit of f(x) as x approaches a is equal to f(a).

Answer

If the limit of f(x) as x approaches a is L, then the limit of f(x)^2 as x approaches a is L^2.

Answer

If the limit of f(x) as x approaches a is L, and the limit of g(x) as x approaches a is 0, then the limit of f(x)/g(x) as x approaches a is L/0.

Question 24

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

Accepted Answer

1

Answer

0

Answer

∞

Question 25

Evaluate the limit of f(x) = (sin(x) + cos(x)) / (sin(x) - cos(x)) as x approaches π/4.

Accepted Answer

1 + √2

Answer

1 - √2

Answer

∞

Answer

0

Question 26

Evaluate the limit of f(x) = (x^3 - 1) / (x - 1) as x approaches infinity.

Accepted Answer

∞

Answer

-∞

Answer

1

Answer

0

Question 27

Determine whether the limit as (x) approaches 0 of ((sin(x))/x)) exists.

Accepted Answer

Yes

Answer

Only if (x) is negative

Answer

Only if (x) is positive

Answer

No

Question 28

Evaluate the limit as (h) approaches 0 of ((f(x+h) - f(x))/h), where (f(x) = x^2 + 1).

Accepted Answer

2x

Answer

x^2

Answer

x

Answer

0

Question 29

Find the limit as (x) approaches 2 of ((sqrt(x^2 - 4))/(x-2)).

Accepted Answer

2

Answer

1

Answer

Does not exist

Answer

-2

Question 30

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

Accepted Answer

The limit of *f(x)* as *x* approaches *a* is *L* if, for every ε > 0, there exists a δ > 0 such that |f(x) - L| < ε whenever 0 < |x - a| < δ.

Answer

The limit of *f(x)* as *x* approaches *a* is *L* if for every ε > 0, there exists a δ > 0 such that |f(x) - L| < δ whenever |x - a| < ε.

Answer

The limit of *f(x)* as *x* approaches *a* is *L* if, for every δ > 0, there exists an ε > 0 such that |f(x) - L| < δ whenever 0 < |x - a| < ε.

Answer

The limit of *f(x)* as *x* approaches *a* is *L* if, for every δ > 0, there exists an ε > 0 such that |f(x) - L| < ε whenever |x - a| < δ.

Question 31

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

Accepted Answer

4

Answer

Undefined

Answer

0

Answer

2

Question 32

Using the Squeeze Theorem, evaluate the limit: lim (x→0) [x² * sin(1/x)]

Accepted Answer

0

Answer

Undefined

Answer

1

Answer

Does not exist

Question 33

Which function has a removable discontinuity at x = 0?

Accepted Answer

f(x) = (x² - 1) / (x - 1)

Answer

f(x) = x²

Answer

f(x) = 1 / x

Answer

f(x) = |x|

Question 34

What is a necessary condition for the existence of a limit of a function *f(x)* as *x* approaches *a*?

Accepted Answer

The left-hand limit and right-hand limit must be equal.

Answer

The function must be defined at x = a.

Answer

The function must be continuous at x = a.

Answer

The function must be differentiable at x = a.

Question 35

Which function has an infinite limit as *x* approaches infinity?

Accepted Answer

f(x) = x² + e^x

Answer

f(x) = 1 / x

Answer

f(x) = cos(x)

Answer

f(x) = sin(x)

Question 36

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

Accepted Answer

4

Answer

8

Answer

2

Answer

0

Question 37

Select the definition that best describes a limit:

Accepted Answer

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

Answer

The instantaneous rate of change of f(x) at x = a.

Answer

The value of the function f(x) at the precise point x = a.

Question 38

Use the epsilon-delta definition to evaluate the limit of f(x) = |x| as x approaches 0.

Accepted Answer

For any ε > 0, choose δ = ε. Then, if |x - 0| < δ, we have |f(x) - 0| = |x| < ε.

Answer

For any ε > 0, choose δ = 1/ε. Then, if |x - 0| < δ, we have |f(x) - 0| = |x| < ε.

Answer

For any ε > 0, choose δ = ε^2. Then, if |x - 0| < δ, we have |f(x) - 0| = |x| < ε.

Question 39

Which theorem allows us to simplify the limit of a sum?

Accepted Answer

The sum rule: lim(f(x) + g(x)) = lim(f(x)) + lim(g(x)).

Answer

The product rule: lim(f(x) * g(x)) = lim(f(x)) * lim(g(x)).

Answer

The quotient rule: lim(f(x) / g(x)) = lim(f(x)) / lim(g(x)).

Question 40

Identify the function that approaches infinity as x approaches negative infinity (-∞):

Accepted Answer

f(x) = x^2

Answer

f(x) = 1 / x

Answer

f(x) = sin(x)

Question 41

Select the appropriate method to evaluate the limit of f(x) = (sqrt(x) - 3) / (x - 9) as x approaches 9.

Accepted Answer

Rationalize the numerator and then use direct substitution.

Answer

Use L'Hopital's rule.

Answer

Apply the squeeze theorem.

Question 42

What type of discontinuity does the function f(x) = 1 / (x - 2) exhibit?

Accepted Answer

Vertical asymptote

Answer

Jump discontinuity

Answer

Removable discontinuity

Question 43

Let f(x) = (x^2 - 4) / (x - 2) for all values of x except x = 2. What is the limit of f(x) as x approaches 2?

Accepted Answer

4

Answer

0

Answer

The limit does not exist.

Answer

2

Question 44

Given that lim_(x→a) f(x) = L and lim_(x→a) g(x) = 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

L - M

Question 45

Find the limit of (sin(x)) / x as x approaches 0.

Accepted Answer

1

Answer

∞

Answer

Does not exist

Answer

0

Question 46

Let f(x) = {x^2 if x < 2, 4 if x = 2, 2x if x > 2}. Which of the following statements is TRUE regarding the limit of f(x) as x approaches 2?

Accepted Answer

The limit does not exist.

Answer

The limit exists and is equal to 4.

Answer

The limit exists and is equal to 3.

Answer

The limit exists and is equal to 2.

Question 47

If lim_(x→a) f(x) = 0 and lim_(x→a) g(x) = ∞, then lim_(x→a) [f(x) * g(x)] equals:

Accepted Answer

The limit may or may not exist, depending on the functions f(x) and g(x).

Answer

∞

Answer

0

Question 48

Find the limit as x approaches 0 of (1 - cos(x)) / x^2.

Accepted Answer

1/2

Answer

1

Answer

∞

Answer

0

Question 49

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

Accepted Answer

If lim_(x→a) f(x) = L, then f(a) must be equal to L.

Answer

The limit of a function at a point may exist even if the function is not defined at that point.

Answer

If lim_(x→a) f(x) = L, then the function f(x) must be defined at x = a.

Question 50

Evaluate lim_(x→∞) (3x^2 + 2x - 1) / (x^2 - 5x + 1).

Accepted Answer

3

Answer

0

Answer

∞

Question 51

Given that lim_(x→2) f(x) = 5 and lim_(x→2) g(x) = -2, find lim_(x→2) [f(x) * g(x)].

Accepted Answer

-10

Answer

3

Answer

7

Answer

10

Question 52

Let f(x) = {x if x ≤ 1, 2x - 1 if x > 1}. Which of the following statements is TRUE regarding the continuity of f(x) at x = 1?

Accepted Answer

f(x) is not continuous at x = 1 because the limit does not exist.

Answer

f(x) is not continuous at x = 1 because f(1) is not defined.

Answer

f(x) is continuous at x = 1.