Limits and Continuity: Interactive Learning Quizzes for Calculus II

Question 1

Is the function f(x) = |x - 5| continuous at x = 5?

Accepted Answer

Yes, because the left and right-hand limits and the function value at x = 5 are equal.

Answer

No, because the left-hand limit does not exist.

Answer

No, because the right-hand limit does not exist.

Answer

Yes, because the function is defined at x = 5.

Question 2

Prove that if f(x) is continuous on a closed interval [a, b], then the range of f(x) is also a closed interval.

Accepted Answer

The Extreme Value Theorem guarantees that f(x) attains its maximum and minimum on [a, b], which form the endpoints of the range, making it closed.

Answer

Boundedness of f(x) on [a, b] does not directly imply that its range is closed.

Answer

Differentiability of f(x) on [a, b] does not guarantee that its range is closed.

Answer

Monotonicity of f(x) on [a, b] does not determine whether its range is closed.

Question 3

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

Accepted Answer

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

Answer

The function is defined at the point.

Question 4

Which of the following functions is NOT continuous on the interval [0, 1]?

Accepted Answer

f(x) = |x - 1|

Answer

f(x) = x^2

Answer

f(x) = sqrt(x)

Question 5

Which of the following theorems can be used to prove that a function is continuous on an interval?

Accepted Answer

The Intermediate Value Theorem

Answer

The Extreme Value Theorem

Question 6

Which of the following is an example of a non-continuous function?

Accepted Answer

f(x) = 1/x

Answer

f(x) = sin(x)

Question 7

Define continuity of a function f(x) at a point a. Explain how each condition in the definition contributes to the concept of continuity.

Accepted Answer

f(a) exists, and the left-hand limit (lim_{x → a-} f(x)) and the right-hand limit (lim_{x → a+} f(x)) both exist and equal f(a).

Answer

The left-hand limit (lim_{x → a-} f(x)) and the right-hand limit (lim_{x → a+} f(x)) exist and are equal, but they may not equal f(a).

Answer

The limit of f(x) as x approaches a exists.

Answer

There exists a number M such that |f(x) - f(a)| < M for all x within a small enough interval around a.

Question 8

Which of the following is a sufficient condition for a function to be continuous on an interval ( [a, b] )?

Accepted Answer

The function is continuous at every point in ( [a, b] ).

Answer

The function is differentiable at every point in ( (a, b) ).

Answer

The function is continuous at the endpoints ( a ) and ( b ).

Answer

The function is monotonic on ( [a, b] ).

Question 9

If ( f(x) ) and ( g(x) ) are continuous on ( [a, b] ) and ( f(x) = g(x) ) for all ( x ) in ( (a, b) ), then what can be concluded about ( f(x) ) and ( g(x) )?

Accepted Answer

( f(x) = g(x) ) for all ( x ) in ( [a, b] ).

Answer

( f'(x) = g'(x) ) for all ( x ) in ( (a, b) ).

Answer

( f''(x) = g''(x) ) for all ( x ) in ( (a, b) ).

Answer

Nothing conclusive can be determined.

Question 10

Which of the following functions is an example of a function that is continuous on ( [0, 1] ) but not differentiable at ( x = 0 )?

Accepted Answer

( f(x) = |x| )

Answer

( f(x) = x^2 )

Answer

( f(x) = rac{1}{x} )

Answer

( f(x) = e^x )

Question 11

Which of the following integrals represents the area under the curve of the continuous function ( f(x) ) on the interval ( [a, b] )?

Accepted Answer

( int_{a}^{b} f(x) dx )

Answer

( int_{b}^{a} f(x) dx )

Answer

( rac{1}{2} int_{a}^{b} f(x) dx )

Answer

( rac{1}{3} int_{a}^{b} f(x) dx )

Question 12

Which of the following is a property of the derivative of a continuous function?

Accepted Answer

The derivative is always continuous.

Answer

The derivative is always differentiable.

Answer

The derivative is always increasing.

Answer

The derivative is always decreasing.

Question 13

Which of the following functions is NOT continuous on its domain?

Accepted Answer

( f(x) = 	an(x) )

Answer

( f(x) = sin(x) )

Answer

( f(x) = cos(x) )

Answer

( f(x) = e^x )

Question 14

Define continuity in terms of limits.

Accepted Answer

lim_(x->a) f(x) = f(a).

Answer

f(a) is defined.

Question 15

At which value of x is the function f(x) = |x - 1| continuous?

Accepted Answer

x = 1

Answer

x = 0

Answer

x = 2

Answer

It is not continuous at any value of x

Question 16

If f(x) is differentiable at x = c, can you definitively conclude that f(x) is continuous at x = c?

Accepted Answer

Yes

Answer

No

Answer

Not enough information provided

Answer

Only if f'(c) = 0

Question 17

For f(x) = {1 if x is rational, 0 if x is irrational}, what is the continuity of f(x) at x = 0?

Accepted Answer

Discontinuous

Answer

Continuous

Answer

Continuous from the left

Answer

Continuous from the right

Question 18

Determine if the function f(x) = |x| is continuous at x = 0.

Accepted Answer

Yes, because the left and right limits of f(x) at x = 0 are equal.

Answer

No, because the function has a sharp corner at x = 0.

Answer

No, because the derivative of f(x) does not exist at x = 0.

Answer

Yes, because the graph of f(x) is smooth at x = 0.

Question 19

Which theorem relates the continuity of a function on an interval to the existence of a derivative?

Accepted Answer

Mean Value Theorem

Answer

Rolle's Theorem

Answer

Extreme Value Theorem

Answer

Intermediate Value Theorem

Question 20

If f(x) and g(x) are both continuous on an interval, which of the following may NOT be continuous on that interval?

Accepted Answer

f(x) / g(x), if g(x) ≠ 0 on the interval

Answer

f(x) + g(x)

Answer

f(x) - g(x)

Answer

f(x) * g(x)

Question 21

State an example of the Intermediate Value Theorem.

Accepted Answer

If f(x) is continuous on an interval, then it takes on every value between its minimum and maximum values.

Answer

If f(x) is differentiable on an interval, then its derivative is continuous on that interval.

Answer

If f(x) has a local maximum at x = c, then f'(c) = 0.

Answer

If f(x) is continuous on an interval, then it has no vertical asymptotes.

Question 22

Which of the following statements is true about a continuous function on a closed interval? Explain why.

Accepted Answer

It attains both its minimum and maximum values on the interval.

Answer

It is differentiable at every point on the interval.

Answer

It is increasing on the entire interval.

Question 23

Which of the following functions is discontinuous at x = 0? Identify the type of discontinuity.

Accepted Answer

f(x) = 1 / sin(x)

Answer

f(x) = x²

Answer

f(x) = cos(x)

Answer

f(x) = e^x

Question 24

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

Accepted Answer

The function is differentiable at all points.

Answer

The function can be approximated by polynomials.

Answer

The function has no breaks or jumps.

Answer

The function can be graphed without lifting the pen.

Question 25

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

Accepted Answer

For any ε > 0, there exists a δ > 0 such that |x - 2| < δ implies |f(x) - f(2)| < ε.

Answer

For any ε > 0, there exists a δ > 0 such that |x - 2| < δ implies |f(x) - f(2)| > ε.

Answer

For any δ > 0, there exists a ε > 0 such that |x - 2| < δ implies |f(x) - f(2)| < ε.

Question 26

Which of the following functions is NOT differentiable?

Accepted Answer

f(x) = |x|

Answer

f(x) = sin(x)

Answer

f(x) = x^2

Question 27

What is a necessary and sufficient condition for a function to be continuous at a point 'a'?

Accepted Answer

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

Answer

The function's graph has no breaks or holes at 'a'.

Answer

The function's first derivative exists at 'a'.

Question 28

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

Accepted Answer

Continuous functions are differentiable everywhere on their domain.

Answer

Continuous functions have no vertical asymptotes.

Answer

Continuous functions are integrable on any closed interval contained in their domain.

Answer

Continuous functions attain both their maximum and minimum values on any closed interval contained in their domain.

Question 29

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

Accepted Answer

If f(x) is continuous on the closed interval [a, b], and f(a) < c < f(b), then there exists a number c in the open interval (a, b) such that f(c) = c.

Answer

If f(x) is continuous on the open interval (a, b), and f'(x) > 0 for all x in (a, b), then f(x) is increasing on (a, b).

Question 30

Which of the following functions is NOT continuous on the entire real line?

Accepted Answer

f(x) = 1/sin(x)

Answer

f(x) = e^x

Answer

f(x) = ln(x)

Answer

f(x) = x^3

Question 31

Which of the following functions is continuous at x = 2?

Accepted Answer

f(x) = x^2

Answer

f(x) = |x - 2|

Answer

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

Answer

f(x) = tan(x - π/2)

Question 32

Which of the following limits does not exist?

Accepted Answer

lim_(x->0) sin(1/x)

Answer

lim_(x->0) e^x

Answer

lim_(x->∞) x/(x+1)

Question 33

The Intermediate Value Theorem states that if f(x) is continuous on [a, b] and y is a number between f(a) and f(b), then there exists a number c in (a, b) such that f(c) = y.

Accepted Answer

True

Answer

Only true if f(x) is increasing

Answer

Only true if f(x) is differentiable

Answer

False

Question 34

If f(x) is continuous on [a, b] and g(x) is continuous on [f(a), f(b)], then the composite function g(f(x)) is continuous on [a, b].

Accepted Answer

True

Answer

False

Answer

Only true if f(x) is differentiable

Question 35

Which of the following theorems guarantees the existence of a root for a continuous function?

Accepted Answer

Intermediate Value Theorem

Answer

Extreme Value Theorem

Answer

Rolle's Theorem

Answer

Maximum Value Theorem

Question 36

Which of the following functions is an example of a piecewise continuous function?

Accepted Answer

f(x) = { 1 if x < 0, 0 if x = 0, -1 if x > 0 }

Answer

f(x) = e^x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Question 37

State the epsilon-delta definition of continuity.

Accepted Answer

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

Answer

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

Answer

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

Question 38

Identify a function that is continuous but not differentiable at x = 0.

Accepted Answer

f(x) = |x|

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 39

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

Accepted Answer

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

Answer

f(x) = sin(x)

Answer

f(x) = x^2 - 4

Answer

f(x) = |x - 2|

Question 40

If a function *f(x)* is continuous on the closed interval [*,a*, *,b*], what can we conclude about the function on that interval?

Accepted Answer

The function *f(x)* attains both its maximum and minimum values within the interval [*,a*, *,b*].

Answer

The function *f(x)* is always positive on the interval [*,a*, *,b*].

Answer

The function *f(x)* is differentiable on the interval [*,a*, *,b*].

Question 41

A function *f(x)* is continuous at *x* = *c*. Which of the following statements MUST be true?

Accepted Answer

The limit of *f(x)* as *x* approaches *c* is equal to *f(c)*:  lim (x->c) *f(x)* = *f(c)*.

Answer

The value of *f(c)* is equal to 0.

Answer

The derivative of *f(x)* exists at *x* = *c*.

Answer

The function *f(x)* is differentiable at *x* = *c*.

Question 42

The function *g(x)* is defined as follows: *g(x)* = { *x*² if *x* < 2; 5 if *x* = 2; 2*x* - 1 if *x* > 2 }. Is *g(x)* continuous at *x* = 2?

Accepted Answer

No, because the left-hand limit and the right-hand limit of *g(x)* as *x* approaches 2 are not equal: lim (x->2-) *g(x)* ≠ lim (x->2+) *g(x)*.

Answer

Yes, because the function is defined for all values of *x*.

Answer

Yes, because *g(2)* is defined.

Question 43

What is the value of lim (x->1) (*x*² - 1) / (*x* - 1)?

Accepted Answer

2

Answer

Does not exist

Answer

0

Answer

1

Question 44

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

Accepted Answer

The sum of two continuous functions is always discontinuous.

Answer

The product of two continuous functions is continuous.

Answer

The composition of two continuous functions is continuous.

Question 45

The Intermediate Value Theorem states that if a function *f(x)* is continuous on the interval [*,a*, *,b*], then for any value *y* between *f(a)* and *f(b)*, there exists a value *c* in [*,a*, *,b*] such that *f(c)* = *y*. What does this theorem guarantee?

Accepted Answer

The existence of at least one solution to the equation *f(x)* = *y*.

Answer

The uniqueness of the solution to the equation *f(x)* = *y*.

Answer

That the function *f(x)* is differentiable on the interval [*,a*, *,b*].

Question 46

Which of the following functions is continuous on the interval (-∞, ∞)?

Accepted Answer

f(x) = e^x

Answer

f(x) = ln(x)

Answer

f(x) = tan(x)

Answer

f(x) = 1/x

Question 47

Suppose *f(x)* is continuous on [*,a*, *,b*] and differentiable on (*a*, *,b*). If *f(a)* = *f(b)*, then according to Rolle's Theorem, there exists a value *c* in (*a*, *,b*) such that:

Accepted Answer

The derivative of *f(x)* at *c* is equal to 0: *f'(c)* = 0.

Answer

The derivative of *f(x)* at *c* is equal to *f(a)*: *f'(c)* = *f(a)*.

Answer

The value of *f(x)* at *c* is equal to 0: *f(c)* = 0.

Answer

The derivative of *f(x)* at *c* is equal to *f(b)*: *f'(c)* = *f(b)*.

Question 48

Which of the following is the precise definition of a function f(x) being continuous at a point x = a?

Accepted Answer

lim_(x→a) f(x) exists, f(a) exists, and lim_(x→a) f(x) = f(a)

Answer

lim_(x→a) f(x) exists

Answer

f(a) exists

Question 49

If f(x) is continuous on the closed interval [a, b] and f(a) < 0 < f(b), which theorem guarantees the existence of a root of f(x) in the interval (a, b)?

Accepted Answer

Intermediate Value Theorem

Answer

Squeeze Theorem

Answer

Mean Value Theorem

Answer

Extreme Value Theorem

Question 50

For what value of k is the function f(x) = {x^2 + 1, x < 2; kx, x ≥ 2} continuous at x = 2?

Accepted Answer

k = 5/2

Answer

No value of k can make the function continuous at x = 2

Answer

k = 2

Answer

k = 1

Question 51

Which of the following types of discontinuities can be removable?

Accepted Answer

Holes

Answer

Vertical Asymptotes

Answer

Oscillating Discontinuities

Answer

Jump Discontinuities

Question 52

If f(x) = |x|, which of the following statements is true?

Accepted Answer

f(x) is continuous everywhere but not differentiable at x = 0

Answer

f(x) is discontinuous at x = 0

Answer

f(x) is differentiable everywhere

Question 53

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

Accepted Answer

1

Answer

Does not exist

Answer

0

Answer

Infinity

Question 54

The function f(x) = 1/x is continuous on the interval:

Accepted Answer

(0, ∞)

Answer

(-∞, ∞)

Answer

[0, ∞)

Answer

(-∞, 0]

Question 55

If f(x) is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f'(c) equals:

Accepted Answer

(f(b) - f(a))/(b - a)

Answer

f(b) - f(a)

Answer

(b - a)/(f(b) - f(a))