Limits and Continuity: Interactive Learning Quizzes for Biology Students

Question 1

Which of the following statements defines a limit?

Accepted Answer

The value that a function approaches as the input approaches a specific value from one side

Answer

The value that a function remains equal to when the input is equal to a specific value

Answer

The value that a function approaches as the input approaches infinity

Answer

The value that a function's rate of change approaches as the input approaches infinity

Question 2

Which of the following theorems can be used to evaluate the limit of the function f(x) = e^(x^2) as x approaches infinity?

Accepted Answer

Squeeze Theorem

Answer

Intermediate Value Theorem

Answer

L'Hopital's Rule

Answer

Mean Value Theorem

Question 3

Which of the following functions is not continuous on the interval [-1, 1]?

Accepted Answer

f(x) = 1 / x

Answer

f(x) = x^2

Answer

f(x) = |x|

Answer

f(x) = e^x

Question 4

Calculate the limit of the sequence {a_n} = (n^2 + 1) / (n^2 - 1) as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

2

Answer

The limit does not exist.

Question 5

Determine the convergence or divergence of the series Σ(n=1 to infinity) (1 / n^2).

Accepted Answer

Convergent

Answer

Divergent

Answer

Cannot be determined

Question 6

Which of the following statements about Taylor series is true?

Accepted Answer

Taylor series can be used to approximate functions with infinitely many derivatives.

Answer

Taylor series can only be used to approximate functions at a single point.

Answer

Taylor series always converge to the function they are approximating.

Answer

Taylor series are only useful for approximating polynomials.

Question 7

When a function has one-sided limits at a point, but the limits are not equal, what can you conclude about the function's continuity at that point?

Accepted Answer

Not continuous

Answer

Continuous

Answer

Cannot be determined

Question 8

Define a limit of a function.

Accepted Answer

The output value that the function approaches as the input value gets arbitrarily close to, but does not equal, a given value.

Answer

The exact output value that the function equals at a specific input value.

Answer

The rate at which the function's output value changes as the input value changes.

Answer

The area enclosed by the function's graph and the x-axis between two given input values.

Question 9

Determine the type of discontinuity of the function f(x) = |x - 1| at x = 1.

Accepted Answer

Jump discontinuity

Answer

Removable discontinuity

Answer

Infinite discontinuity

Answer

Continuous

Question 10

Which theorem can be used to prove the continuity of a rational function?

Accepted Answer

Quotient Rule

Answer

Sum/Difference Rule

Answer

Product Rule

Answer

Chain Rule

Question 11

Find the limit of the sequence a_n = (n^2 + 1) / (n^2 - 1) as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

Infinity

Answer

The limit does not exist.

Question 12

Evaluate the limit of the function f(x) = (cos(x) - 1) / x as x approaches 0.

Accepted Answer

0

Answer

1

Answer

-1

Answer

The limit does not exist.

Question 13

Determine if the function m(x) = { x^2 if x < 2, 4 - x if x ≥ 2 } is continuous at x = 2.

Accepted Answer

Discontinuous at x = 2 because the left and right limits are not equal.

Answer

Continuous at x = 2 because the left and right limits are equal.

Answer

Continuous at x = 2 because the function is defined at x = 2.

Answer

Discontinuous at x = 2 because the function is not differentiable at x = 2.

Question 14

Consider the function s(x) = √x. Determine the intervals where s(x) is continuous.

Accepted Answer

(0, ∞)

Answer

[0, ∞)

Answer

(-∞, 0]

Answer

(-∞, ∞)

Question 15

A limit is the value that a function approaches as the input approaches a specific value.

Accepted Answer

True

Answer

False

Question 16

A necessary condition for a function to be continuous at a point is that the limit of the function at that point exists.

Accepted Answer

True

Answer

False

Question 17

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

Accepted Answer

True

Answer

False

Question 18

The order of the removable discontinuity of the function f(x) = (x^2 - 1) / (x - 1) at x = 0 is 1.

Accepted Answer

True

Answer

False

Question 19

The function f(x) = 1 / x has a jump discontinuity at x = 0.

Accepted Answer

True

Answer

False

Question 20

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

Accepted Answer

The derivative of the function exists at that point.

Answer

The limit of the function exists at that point.

Answer

The function is defined at that point.

Question 21

Which of the following theorems guarantees the existence of a limit?

Accepted Answer

Bolzano-Weierstrass Theorem

Answer

Squeeze Theorem

Answer

Intermediate Value Theorem

Answer

Extreme Value Theorem

Question 22

Determine whether the function (f(x) = |x - 3|) is continuous at (x = 3).

Accepted Answer

Discontinuous due to a jump discontinuity

Answer

Continuous

Answer

Discontinuous due to an infinite discontinuity

Answer

Discontinuous due to a removable discontinuity

Question 23

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

Accepted Answer

They are differentiable at every point.

Answer

They have no vertical asymptotes.

Answer

They can be integrated over closed intervals.

Answer

They can have removable discontinuities.

Question 24

Use the epsilon-delta definition of continuity to prove that the function (f(x) = x^2) is continuous at (x = 1).

Accepted Answer

Let (varepsilon > 0), then choose (delta = sqrt{varepsilon}).

Answer

Let (varepsilon > 0), then choose (delta = varepsilon/2).

Answer

Let (varepsilon > 0), then choose (delta = 1/varepsilon).

Answer

Cannot be proven with the given definition.

Question 25

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

Accepted Answer

0

Answer

1

Answer

1/2

Answer

Limit does not exist

Question 26

Which of the following is NOT a characteristic of a continuous function?

Accepted Answer

The derivative of the function is continuous

Answer

The limit of the function at a point exists

Answer

The function is defined at all points in its domain

Answer

The graph of the function has no breaks

Question 27

Explain the epsilon-delta definition of a limit. Use clear and concise language, avoiding technical jargon.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 28

Determine the correct formula for the derivative of f(x) = sin(x^2). Provide a brief explanation of your choice.

Accepted Answer

2x cos(x^2)

Answer

cos(x^2)

Answer

sin(2x)

Answer

2x^2 cos(x^2)

Question 29

In mathematics, a limit of a function is the value that the function approaches as the input value approaches a specific number. Which of the following is the formal definition of a limit?

Accepted Answer

lim_(x->a) f(x) = L if for any number ε>0, there exists a number 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'(a) = L.

Answer

lim_(x->a) f(x) = L if the graph of f(x) has a vertical asymptote at x = a.

Question 30

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

Accepted Answer

lim_(x->c) f(x) exists.

Answer

f(c) is defined.

Answer

f'(c) exists.

Answer

f''(c) exists.

Question 31

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

Accepted Answer

0

Answer

1

Answer

2

Answer

Does not exist

Question 32

Which of the following is a graphical representation of a continuous function?

Accepted Answer

A smooth curve with no breaks or sharp corners

Answer

A curve with sharp corners

Answer

A curve with vertical asymptotes

Answer

A curve with horizontal asymptotes

Question 33

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

Accepted Answer

0

Answer

1

Answer

Does not exist

Question 34

Find the derivative of the function f(x) = x^3 - 2x^2 + 1.

Accepted Answer

3x^2 - 4x

Answer

3x^2 - 2

Answer

x^3 - 4x

Answer

x^3 - 2x + 1

Question 35

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

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 36

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

Accepted Answer

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

Answer

The graph of the function has no holes or jumps at a.

Answer

The derivative of the function exists at a.

Question 37

Which of the following functions is discontinuous at x = 0?

Accepted Answer

f(x) = |x| / x

Answer

f(x) = sin(x)

Answer

f(x) = x^2

Answer

f(x) = e^x

Question 38

Determine the limit of the function f(x) = (1 - cos(x)) / x^2 as x approaches 0.

Accepted Answer

1/2

Answer

Does not exist

Answer

-1/2

Answer

0

Question 39

A function is continuous at a point if the following conditions hold true. Which option accurately describes these conditions?

Accepted Answer

The function's limit as *x* approaches the point exists, and this limit is equal to the function's value at that point.

Answer

The function has a defined value at that point.

Answer

The function is differentiable at that point.

Answer

The function's limit as *x* approaches the point exists, but it doesn't have to equal the function's value at that point.

Question 40

What is the limit of the function *f(x) = (sin(x) - x) / (x^3 - x)* as *x* approaches 0?

Accepted Answer

0

Answer

-1

Answer

1

Answer

Does not exist

Question 41

Which of the following functions is continuous over the entire real number line (*(-∞, ∞)*)?

Accepted Answer

f(x) = e^x

Answer

f(x) = sqrt(x)

Answer

f(x) = 1 / x

Answer

f(x) = tan(x)

Question 42

The function *f(x) = 1 / (x - 2)* exhibits a discontinuity at *x = 2*. What type of discontinuity is this?

Accepted Answer

Removable discontinuity

Answer

Essential discontinuity

Answer

Jump discontinuity

Answer

Infinite discontinuity

Question 43

Evaluate the limit of *(x^2 - 4) / (x - 2)* as *x* approaches 2.

Accepted Answer

4

Answer

0

Answer

6

Answer

Cannot be determined

Question 44

Is the function *f(x) = 1 / (x - 3)* continuous at *x = 3*?

Accepted Answer

Not continuous

Answer

Continuous from the left but not from the right

Answer

Continuous from the right but not from the left

Answer

Continuous

Question 45

What is the horizontal asymptote of the function *g(x) = (x^2 - 1) / (x^2 + 1)*?

Accepted Answer

y = 1

Answer

y = -1

Answer

No horizontal asymptote

Answer

y = 0

Question 46

Using the limit definition of the derivative, what is the derivative of *f(x) = x^3* at *x = 1*?

Accepted Answer

3

Answer

Cannot be determined

Answer

1

Answer

0

Question 47

Is the function *h(x) = |x - 1|* differentiable at *x = 1*?

Accepted Answer

Not differentiable

Answer

Differentiable

Answer

Differentiable from the right but not from the left

Answer

Differentiable from the left but not from the right

Question 48

What is the limit of *(e^x - 1) / x* as *x* approaches 0?

Accepted Answer

1

Answer

e

Answer

0

Answer

Does not exist

Question 49

Determine the continuity of the function k(x) = 1 / (sin(x)) at x = 0.

Accepted Answer

Not continuous

Answer

Continuous from the right but not from the left

Answer

Continuous from the left but not from the right

Answer

Continuous

Question 50

Use the squeeze theorem to evaluate the limit of (x^2 - 1) / (x^2 + 1) as x approaches 1.

Accepted Answer

0

Answer

-1

Answer

1

Answer

Cannot be determined

Question 51

Determine the number that the function f(x) = x^2 - 4 approaches as x becomes very large.

Accepted Answer

∞

Answer

4

Answer

Cannot be determined

Answer

-∞

Question 52

Find the vertical asymptote(s) of the function g(x) = (x + 1) / (x^2 - 1).

Accepted Answer

x = 1, x = -1

Answer

No vertical asymptotes

Answer

x = 2

Answer

x = 0