Limits and Continuity: Master the Fundamentals of Calculus

Question 1

For the function f(x) = (x^2 - 1) / (x - 1), what type of discontinuity does it have at x = 1?

Accepted Answer

Removable

Answer

Continuous

Answer

Infinite

Answer

None

Question 2

Which of the following is NOT a necessary condition for a function to be continuous at a point?

Accepted Answer

The function must be differentiable at that point.

Answer

The function must be defined at that point.

Answer

The limit of the function at that point exists.

Answer

The graph of the function must have no hole or jump at that point.

Question 3

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

Accepted Answer

∞

Answer

-∞

Answer

-1

Answer

0

Question 4

Can a function have a continuous graph but not be differentiable at a particular point?

Accepted Answer

Yes

Answer

No

Question 5

Which of the following statements is true about the function $$f(x) = e^{2x}$$? $$

Accepted Answer

It is continuous and differentiable everywhere.

Answer

It is continuous but not differentiable at $$x = 0$$.

Answer

It is differentiable but not continuous at $$x = 0$$.

Answer

It is neither continuous nor differentiable at $$x = 0$$. $$

Question 6

The limit of $$\frac{\sin(x)}{x}$$ as $$x$$ approaches 0 is:

Accepted Answer

0

Answer

1

Answer

-1

Answer

The limit does not exist.

Question 7

Find the domain of the function $$f(x) = \sqrt{x - 2}$$.

Accepted Answer

$$x \ge 2$$

Answer

$$x \le 2$$

Answer

$$x > 2$$

Answer

All real numbers

Question 8

Which of the following is NOT a property of continuous functions?

Accepted Answer

The function is differentiable at every point.

Answer

The graph has no breaks or jumps.

Answer

The limit as $$x$$ approaches any value exists.

Answer

The function is integrable on any closed interval.

Question 9

Is it possible for a function to have removable discontinuities at multiple points?

Accepted Answer

Yes, a function can have multiple removable discontinuities.

Answer

No, a function can only have one removable discontinuity.

Question 10

Which of the following is the most precise definition of a limit?

Accepted Answer

As the input approaches a specific value, the function approaches a specific number.

Answer

The number that the function equals when the input equals a specific value.

Answer

The rate of change of the function at a specific point.

Answer

The slope of the tangent line to the graph of the function at a specific point.

Question 11

Which function below is not continuous at x = 0?

Accepted Answer

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

Answer

f(x) = x^2

Answer

f(x) = |x|

Answer

f(x) = e^x

Question 12

Define the limit of a function f(x) as x approaches a value c.

Accepted Answer

The value L such that for any positive number ε, there exists a positive number δ such that whenever 0 < |x - c| < δ, then |f(x) - L| < ε.

Answer

The value of f(x) when x equals c.

Answer

The derivative of f(x) evaluated at c.

Question 13

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

Accepted Answer

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

Answer

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

Answer

f(x) = sin(x)

Answer

f(x) = |x|

Question 14

Which of the following techniques can be used to evaluate the limit of a function that is indeterminate in the form 0/0 or ∞/∞?

Accepted Answer

L'Hôpital's Rule

Answer

Product Rule

Answer

Quotient Rule

Answer

Chain Rule

Question 15

Which of the following is a necessary condition for the Intermediate Value Theorem to apply to a function f(x) on an interval [a, b]?

Accepted Answer

f(x) is continuous on [a, b].

Answer

f(x) is differentiable on [a, b].

Answer

f(x) is increasing on [a, b].

Answer

f(x) is concave up on [a, b].

Question 16

Which of the following is a consequence of the Extreme Value Theorem?

Accepted Answer

A continuous function on a closed interval [a, b] attains both a minimum and a maximum value.

Answer

A differentiable function on a closed interval has at least one critical point.

Answer

An increasing function on a closed interval is bounded.

Question 17

Which of the following tests can be used to determine whether a function f(x) is uniformly continuous on an interval [a, b]?

Accepted Answer

Cauchy Test

Answer

Ratio Test

Answer

Root Test

Answer

Comparison Test

Question 18

Identify the horizontal asymptote of the function f(x) = (3x^2 - 2x + 1)/(x^2 - 1) if it exists.

Accepted Answer

y = 3

Answer

y = -3

Answer

y = 2

Answer

The function has no horizontal asymptotes

Question 19

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

Accepted Answer

x = 2

Answer

The function has no vertical asymptotes.

Question 20

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

Accepted Answer

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

Answer

f(x) = |x - 2|

Answer

f(x) = tan x

Answer

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

Question 21

If f(x) is a continuous function and lim x->a f(x) = L, then which of the following is always true?

Accepted Answer

f(a) = L

Answer

f(a) is undefined

Answer

f(a) is always negative

Answer

f(a) is always positive

Question 22

Provide the epsilon-delta definition of the limit lim x->2 f(x) = 5, where f(x) is a function defined on the real numbers.

Accepted Answer

For every epsilon > 0, there exists a delta > 0 such that if 0 < |x - 2| < delta, then |f(x) - 5| < epsilon

Answer

For every epsilon > 0, there exists a delta > 0 such that if x > 2, then f(x) < epsilon

Question 23

Suppose that the function k(x) is continuous on the closed interval [a, b]. Which of the following statements accurately describes the graph of k(x)?

Accepted Answer

It has no sudden jumps or breaks in the interval.

Answer

It may oscillate rapidly and unpredictably.

Answer

It may have sharp corners or cusps.

Answer

It may have infinitely many points.

Question 24

Let f(x) and g(x) be continuous functions. Which of the following functions is not necessarily continuous?

Accepted Answer

f(x) / g(x)

Answer

f(g(x))

Answer

f(x) + g(x)

Question 25

If the function m(x) is continuous on the closed interval [0, 1] and m(0) = 2 and m(1) = 5, what can be concluded about m(x)?

Accepted Answer

It takes on every value in the range [2, 5] at some point in the interval.

Answer

It takes on the value 0 at some point in the interval.

Answer

It has at least one maximum and one minimum in the interval.

Answer

It is strictly increasing on the entire interval.

Question 26

Which of the following functions has an infinite discontinuity at x = 0?

Accepted Answer

1 / x

Answer

x^2

Answer

e^x

Answer

sqrt(x)

Question 27

Which of the following statements is not a necessary condition for a function to be continuous on its domain?

Accepted Answer

It is monotone increasing.

Answer

It is differentiable at every point in its domain.

Answer

It has a finite number of discontinuities.

Question 28

Evaluate the following limit using the epsilon-delta definition of a limit: lim (x^3 - 1) / (x - 1) as x approaches 1

Accepted Answer

Given any ε > 0, there exists δ > 0 such that if 0 < |x - 1| < δ, then |(x^3 - 1) / (x - 1) - 3| < ε

Answer

Given any ε > 0, there exists δ > 0 such that if |x - 1| > δ, then |(x^3 - 1) / (x - 1) - 3| < ε

Answer

Given any δ > 0, there exists ε > 0 such that if 0 < |x - 1| < δ, then |(x^3 - 1) / (x - 1) - 3| < ε

Question 29

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

Accepted Answer

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

Answer

lim (x->a) f(x) = L if f(a) = L.

Answer

lim (x->a) f(x) = L if there is an open interval containing a such that, for all x in that interval except possibly a, f(x) = L.

Question 30

Which of the following consequences stem from the Intermediate Value Theorem for continuous functions?

Accepted Answer

If a function is continuous on a closed interval and takes on two different values at the endpoints, then it must take on every value between those two values at some point in the interval.

Answer

If a function is continuous on an open interval, then it is differentiable on that interval.

Answer

If a function is continuous on a closed interval, then it must have a maximum and minimum value on that interval.

Question 31

Which of the following statements about the function g(x) = 1 / (x - 2) is accurate?

Accepted Answer

It is continuous on its domain which is all real numbers except x = 2.

Answer

It is continuous at x = 2.

Answer

It is continuous on the entire real line.

Question 32

Identify the function from the following choices that is not continuous at x = 0:

Accepted Answer

f(x) = |x - 1|

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 33

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

Accepted Answer

The function is continuous at that point.

Answer

The function is increasing at that point.

Answer

The function has a local maximum at that point.

Answer

The function has a derivative at that point.

Question 34

Which of the following 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 if 0 < |x - a| < δ, then |f(x) - L| < ε

Answer

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

Answer

lim_(x->a) f(x) = L if f(x) is defined at a

Question 35

Determine whether the following function is continuous at x = 1:
f(x) = {x^2 + 1 if x ≠ 1
{2 if x = 1

Accepted Answer

Discontinuous

Answer

Continuous

Answer

Cannot be determined

Question 36

Which of the following is a necessary condition for the existence of a horizontal asymptote?

Accepted Answer

lim_(x->∞) f(x) exists

Answer

lim_(x->∞) f''(x) exists

Answer

lim_(x->∞) f(x) = 0

Answer

lim_(x->∞) f'(x) exists

Question 37

Which of the following is an application of limits and continuity in real-world problems?

Accepted Answer

Calculating the instantaneous rate of change of a moving object

Answer

Predicting the weather

Answer

Finding the area of a circle

Answer

Solving trigonometric equations

Question 38

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

Accepted Answer

2

Answer

1

Answer

Does not exist

Answer

0

Question 39

Determine the continuity of the function f(x) = |x| at x = 0.

Accepted Answer

Not continuous

Answer

Right-hand continuous only

Answer

Continuous

Answer

Left-hand continuous only

Question 40

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

Accepted Answer

1

Answer

0

Answer

Does not exist

Question 41

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

Accepted Answer

e

Answer

0

Answer

Does not exist

Answer

1

Question 42

Determine the continuity of the function f(x) = x^2 + 1 on the interval (-∞, ∞).

Accepted Answer

Continuous

Answer

Right-hand continuous only

Answer

Left-hand continuous only

Answer

Not continuous

Question 43

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

Accepted Answer

1/2

Answer

Does not exist

Answer

-1/2

Answer

0

Question 44

Determine the continuity of the function f(x) = 1/(x - 1) on the interval (0, 2).

Accepted Answer

Not continuous

Answer

Right-hand continuous only

Answer

Continuous

Answer

Left-hand continuous only

Question 45

Evaluate the limit of the function f(x) = (sqrt(x + 2) - sqrt(2))/x as x approaches 0.

Accepted Answer

1/(2 * sqrt(2))

Answer

Does not exist

Answer

-1/(2 * sqrt(2))

Answer

0

Question 46

Determine the continuity of the function f(x) = |x + 1| at x = -1.

Accepted Answer

Not continuous

Answer

Continuous

Answer

Left-hand continuous only

Question 47

Find the limit of the function f(x) = (x^3 - 8)/(x - 2) as x approaches 2.

Accepted Answer

16

Answer

0

Answer

-16

Answer

Does not exist