Limits and Continuity: Master Calculus Fundamentals

Question 1

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

Accepted Answer

1

Answer

0

Answer

π/2

Answer

Undefined

Question 2

Use the epsilon-delta definition of a limit to prove that lim (x^2 - 1) / (x - 1) as x -> 1 = 2.

Accepted Answer

Let ε > 0. Then there exists a δ > 0 such that 0 < |x - 1| < δ implies |[(x^2 - 1) / (x - 1)] - 2| < ε.

Answer

Let ε > 0. Then there exists a δ > 0 such that |x - 1| < δ implies |[(x^2 - 1) / (x - 1)] - 2| < ε.

Answer

Let δ > 0. Then there exists an ε > 0 such that 0 < |x - 1| < δ implies |[(x^2 - 1) / (x - 1)] - 2| < ε.

Answer

Let δ > 0. Then there exists an ε > 0 such that |x - 1| < δ implies |[(x^2 - 1) / (x - 1)] - 2| > ε.

Question 3

Which of the following statements about the function f(x) = 1 / (x^2 - 4) is true?

Accepted Answer

f(x) has a vertical asymptote at x = 2.

Answer

f(x) is continuous at x = 2.

Answer

f(x) has a removable discontinuity at x = 2.

Answer

f(x) is continuous everywhere.

Question 4

State a consequence of a function being continuous on an interval

Accepted Answer

The function attains both a maximum and minimum value on the interval.

Answer

The function has no zeros on the interval.

Answer

The graph of the function is a straight line on the interval.

Answer

The function is differentiable on the interval.

Question 5

Which function is discontinuous at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 6

Find the slope of the tangent line to the graph of f(x) = x^3 - 2x^2 at x = 1.

Accepted Answer

3

Answer

-1

Answer

1

Answer

5

Question 7

Which condition is necessary for a function to be differentiable at a point?

Accepted Answer

The function must be continuous at that point.

Answer

The function must have a derivative at that point.

Answer

The function must be increasing at that point.

Answer

The function must be decreasing at that point.

Question 8

Which function is continuous but not differentiable at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 9

Which of the following limits is indeterminate?

Accepted Answer

lim_(x->0) (x^2 + 1) / (x - 1)

Answer

lim_(x->2) (x^3 - 8) / (x - 2)

Answer

lim_(x->3) (x - 3)^2 / (x - 3)

Answer

lim_(x->4) (x^2 - 16) / (x - 4)

Question 10

Which of the following functions has a discontinuity at x = 1?

Accepted Answer

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

Answer

f(x) = x^2

Answer

f(x) = |x - 1|

Answer

f(x) = sqrt(x - 1)

Question 11

Which of the following is the correct definition of continuity at a point?

Accepted Answer

The limit of the function as x approaches the point exists and is equal to the function value at the point

Answer

The function is defined at the point

Answer

The derivative of the function exists at the point

Answer

The function has a vertical asymptote at the point

Question 12

Apply l'Hôpital's rule to evaluate the limit of the function f(x) = (e^x - 1 - x)/x as x approaches 0.

Accepted Answer

1

Answer

0

Answer

e

Answer

∞

Question 13

Which of the following best defines the limit of a function?

Accepted Answer

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

Answer

The slope of a function at a specific point on its graph

Answer

The rate at which a function changes at a specific point on its graph

Answer

The maximum or minimum value of a function

Question 14

Which of the following graphs represents a function that is not continuous at x = 0?

Accepted Answer

A graph with a hole at x = 0

Answer

A straight line passing through the origin

Answer

A parabola opening upwards

Answer

A circle centered at the origin

Question 15

A function is said to have a removable discontinuity at a point if:

Accepted Answer

Its limit exists at that point but the function is not defined at that point

Answer

Its limit does not exist at that point

Answer

Its graph has a hole at that point

Answer

Its derivative does not exist at that point

Question 16

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

Accepted Answer

e^2

Answer

0

Answer

1

Answer

Infinity

Question 17

Which function has an infinite discontinuity at x = 2?

Accepted Answer

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

Answer

f(x) = |x - 2|

Answer

f(x) = sin(x - 2)

Answer

f(x) = e^(x - 2)

Question 18

Describe the relationship between the limit of a sequence and the limit of the corresponding function.

Accepted Answer

The limit of the sequence is equal to the limit of the function at the limit point of the sequence.

Answer

The limit of the sequence is always equal to the limit of the function at any point.

Answer

The limit of the function is always equal to the limit of the sequence.

Answer

There is no relationship between the limit of a sequence and the limit of the corresponding function.

Question 19

Evaluate the limit using L'Hopital's Rule: lim x->0 (sin(x) - x) / (x^3 - x)

Accepted Answer

0

Answer

1

Answer

Limit does not exist

Question 20

Determine if the function f(x) = { x^2, if x ≤ 0; x - 1, if x > 0 } is continuous at x = 0. Explain your reasoning.

Accepted Answer

Continuous at x = 0: Left and right-hand limits are equal.

Answer

Not continuous at x = 0: Left and right-hand limits differ.

Answer

Continuous at x = 0: Limit of the function exists.

Answer

Not continuous at x = 0: Function is not differentiable at x = 0.

Question 21

What is the connection between the Intermediate Value Theorem and the continuity of a function f(x) on an interval [a, b]?

Accepted Answer

If f(x) is continuous on [a, b], then the Intermediate Value Theorem applies to f(x) on [a, b].

Answer

If f(x) is discontinuous at a point c in [a, b], then the Intermediate Value Theorem does not apply to f(x) on [a, b].

Answer

If f(x) is continuous on [a, b], then the Intermediate Value Theorem guarantees that f(x) has a root in [a, b].

Question 22

The function f(x) = |x - 1| is defined as the absolute value of (x - 1).  What happens to f(x) as x gets closer and closer to 1?

Accepted Answer

f(x) approaches 1.

Answer

f(1) is undefined.

Answer

f(x) is differentiable at x = 1, so it is continuous there.

Question 23

Which value of x makes the function f(x) = (x^2 - 4) / (x - 2) undefined?

Accepted Answer

x = 2

Answer

x = 0

Answer

x = 4

Answer

x = -4

Question 24

Identify the function that is not differentiable at x = 0.

Accepted Answer

f(x) = |x|

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Answer

f(x) = x^2

Question 25

Find the equation of the horizontal asymptote of the function f(x) = (2x^3 - 5x + 1) / (x^2 - 4).

Accepted Answer

y = 2

Answer

y = 0

Answer

y = 4

Answer

y = 5

Question 26

Evaluate the limit: lim_(x→∞) (√(x^2 + 1) - x).

Accepted Answer

0

Answer

1

Answer

-1

Answer

∞

Question 27

Determine whether the function f(x) = { 1/x if x ≠ 0, 0 if x = 0 } has an infinite discontinuity at x = 0.

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Question 28

Which of the following limits is equivalent to the derivative of the function f(x) at x = 2?

Accepted Answer

lim_(h→0) [f(2 + h) - f(2)] / h

Answer

lim_(h→∞) [f(2 + h) - f(2)] / h

Answer

lim_(x→2) f(x) / x

Answer

lim_(x→∞) f(x) / f(2)

Question 29

Determine whether the following statement is true or false: A function that is continuous at a point must be differentiable at that point.

Accepted Answer

False

Answer

Cannot be determined

Answer

True

Question 30

Identify the type of discontinuity exhibited by the function f(x) = 1/x at x = 0.

Accepted Answer

Infinite Discontinuity

Answer

Jump Discontinuity

Answer

Removable Discontinuity

Answer

Essential Discontinuity

Question 31

State the conditions that must be satisfied for a function to be continuous at a point x = a.

Accepted Answer

The function must be defined at x = a, the limit as x approaches a must exist, and the limit must be equal to the function's value at x = a.

Answer

The function must be differentiable at x = a.

Question 32

If the left-hand limit and right-hand limit of f(x) at x = a are not equal, what can be concluded about the continuity of f(x) at x = a?

Accepted Answer

f(x) is not continuous at x = a.

Answer

f(x) is differentiable at x = a.

Answer

Nothing can be concluded about the continuity of f(x) at x = a.

Answer

f(x) is continuous at x = a.

Question 33

Which statement correctly describes the limit of a function as x approaches a certain value?

Accepted Answer

The limit exists only when the left-hand limit and the right-hand limit are equal.

Answer

The limit is always equal to the function's value at that specific value of x.

Answer

The limit exists only if the function is defined at that specific value of x.

Answer

The limit always exists for any function.

Question 34

The function f(x) = |x| has a discontinuity at x = 0. What type of discontinuity is it?

Accepted Answer

Jump discontinuity

Answer

Infinite discontinuity

Answer

Oscillating discontinuity

Answer

Removable discontinuity

Question 35

Which function is continuous for all real numbers?

Accepted Answer

f(x) = x³ + 2x - 1

Answer

f(x) = |x - 2|

Answer

f(x) = 1 / x

Answer

f(x) = tan(x)

Question 36

Evaluate the limit of the function f(x) = (x^2 - 1)/(x - 1) as x approaches 1.

Accepted Answer

2

Answer

Does not exist

Answer

0

Answer

1

Question 37

Which condition ensures that a function f(x) is continuous at a point 'a'?

Accepted Answer

The function must be defined, finite, and its limit as x approaches 'a' must exist and be equal to f(a).

Answer

The function must have a vertical asymptote at that point.

Answer

The function must be a polynomial.

Answer

The function must be differentiable at that point.

Question 38

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

Accepted Answer

1

Answer

Does not exist

Answer

∞

Answer

0

Question 39

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

Accepted Answer

0

Answer

1

Answer

Does not exist

Answer

∞

Question 40

If the limit of a function f(x) as x approaches 'a' exists, what can be concluded about the function's continuity at that point?

Accepted Answer

The function may or may not be continuous at that point.

Answer

The function is discontinuous at that point.

Answer

The function is continuous at that point.

Question 41

Evaluate the limit as x approaches 2 from the left of the function f(x) = 1/(x - 2).

Accepted Answer

-∞

Answer

∞

Answer

0

Answer

1

Question 42

Which of the following functions is continuous at x = -1?

Accepted Answer

f(x) = x^2 - 1

Answer

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

Answer

f(x) = |x + 1|

Answer

f(x) = 1/x

Question 43

At what point does the function f(x) = (x^2 - 9)/(x - 3) have a removable discontinuity?

Accepted Answer

x = 3

Answer

The function has no removable discontinuity.

Answer

x = 0

Answer

x = -3