Definition of a Limit: A Comprehensive Guide for Calculus Students

Question 1

In Calculus, what is the formal definition of a limit?

Accepted Answer

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

Answer

f(a) = L

Answer

f(x) is defined at a

Answer

f(x) approaches L as x approaches a

Question 2

Which statement accurately reflects the epsilon-delta definition of a limit?

Accepted Answer

If |x - a| is less than delta, then |f(x) - L| is less than epsilon.

Answer

If |x - a| is greater than delta, then |f(x) - L| is greater than epsilon.

Answer

If |x - a| is less than delta, then |f(x) - L| is greater than epsilon.

Answer

If |x - a| is greater than delta, then |f(x) - L| is less than epsilon.

Question 3

Using the epsilon-delta definition, describe the limit of the function f(x) = x² - 4 as x approaches 2.

Accepted Answer

For any ε > 0, there exists a δ > 0 such that if 0 < |x - 2| < δ, then |x² - 4| < ε.

Answer

For any δ > 0, there exists an ε > 0 such that if 0 < |x - 2| < δ, then |x² - 4| > ε.

Answer

For any ε > 0, there exists a δ > 0 such that if |x - 2| > δ, then |x² - 4| < ε.

Answer

For any δ > 0, there exists an ε > 0 such that if |x - 2| < δ, then |x² - 4| < ε.

Question 4

Which of the following is a divergent sequence?

Accepted Answer

{n}

Answer

{1 / n^2}

Answer

{sin(n)}

Answer

{(-1)^n}

Question 5

Find the limit of the function f(x) = (sin(x) + 1) / (x + 1) as x approaches infinity.

Accepted Answer

1

Answer

0

Answer

-1

Answer

The limit does not exist.

Question 6

Which of the following is a vertical asymptote of the function f(x) = x / (x - 2)?

Accepted Answer

x = 2

Answer

x = -2

Answer

x = 0

Answer

The function has no vertical asymptotes.

Question 7

If (\lim_{x 	o c} f(x) = L) and (\lim_{x 	o c} g(x) = M), then (\lim_{x 	o c} [f(x) + g(x)] = )

Accepted Answer

L + M

Answer

0

Answer

L - M

Answer

M - L

Question 8

What is the graphical interpretation of a limit?

Accepted Answer

As (x) approaches (c), the graph of (f(x)) gets closer and closer to the line (y = L), without necessarily touching it.

Answer

As (x) approaches (c), the graph of (f(x)) intersects the line (y = L).

Answer

As (x) approaches (c), the graph of (f(x)) becomes vertical.

Answer

As (x) approaches (c), the graph of (f(x)) disappears.

Question 9

Which of the following is not a technique for evaluating limits?

Accepted Answer

Integration

Answer

Substitution

Answer

Factoring

Answer

Rationalization

Question 10

If (\lim_{x 	o c} f(x) = L) and (\lim_{x 	o c} g(x) = M), then (\lim_{x 	o c} [f(x) - g(x)] = )

Accepted Answer

L - M

Answer

0

Answer

M - L

Answer

L + M

Question 11

What is the interpretation of the limit of a function f(x) as x approaches c being equal to infinity?

Accepted Answer

As x gets closer and closer to c, the values of f(x) grow without bound.

Answer

As x gets closer and closer to c, the values of f(x) approach a specific finite value.

Answer

As x gets closer and closer to c, the values of f(x) oscillate between two finite values.

Answer

The function f(x) has a hole at x = c.

Question 12

What is the limit of the sequence {n^2 + 1} as n approaches infinity?

Accepted Answer

∞

Answer

0

Answer

1

Answer

Does not exist

Question 13

State the Squeeze Theorem.

Accepted Answer

If f(x) ≤ g(x) ≤ h(x) for all x in an open interval containing c, and if 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) = L and 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)] = LM.

Answer

If lim x→c f(x) = L and lim x→c g(x) does not exist, then lim x→c [f(x)g(x)] does not exist.

Question 14

Which of the following is the formal definition of a limit?

Accepted Answer

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

Answer

The value of the function at the point x = c

Answer

The slope of the function at the point x = c

Answer

The area under the curve of the function from x = c to x = a

Question 15

What does it mean for a limit to exist?

Accepted Answer

The output of the function approaches a specific value as the input approaches a specific value

Answer

The function is continuous at the point where the limit is being taken

Answer

The function has a derivative at the point where the limit is being taken

Answer

The function is increasing or decreasing without bound as the input approaches a specific value

Question 16

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

Accepted Answer

0

Answer

1

Answer

-1

Answer

Does not exist

Question 17

What is the limit of (1 / x) as x approaches infinity?

Accepted Answer

0

Answer

1

Answer

Infinity

Answer

-Infinity

Question 18

Which of the following expressions represents the formal definition of a limit?

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

Answer

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

Answer

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

Question 19

Which property of limits states that the limit of a sum is the sum of the limits?

Accepted Answer

Limit Law of Sums

Answer

Limit Law of Products

Answer

Limit Law of Quotients

Answer

Cauchy's Mean Value Theorem

Question 20

Which theorem provides a sufficient condition for a function to be continuous at a point?

Accepted Answer

Intermediate Value Theorem

Answer

Extreme Value Theorem

Answer

Cauchy's Mean Value Theorem

Answer

Mean Value Theorem

Question 21

What is the graphical interpretation of a limit?

Accepted Answer

The value that the function approaches on the graph as the input approaches a certain value

Answer

The point where the function crosses the x-axis

Answer

The slope of the tangent line to the graph at a certain point

Answer

The area under the curve of the function between two points

Question 22

Which method is used to evaluate limits that involve indeterminate forms?

Accepted Answer

L'Hopital's Rule

Answer

Chain Rule

Answer

Product Rule

Answer

Quotient Rule

Question 23

Which of the following statements about limits is true?

Accepted Answer

Limits can exist even if the function is discontinuous at the input value.

Answer

Limits can only exist for finite values of the input.

Answer

Limits can only exist for continuous functions.

Answer

Limits are always unique.

Question 24

Which of the following is the formal definition of a limit?

Accepted Answer

The limit of a function f(x) as x approaches a is the value L such that, for any positive number ε, there exists a positive number δ such that if 0 < |x - a| < δ, then |f(x) - L| < ε.

Answer

The limit of a function f(x) as x approaches a is the value of f(a).

Answer

The limit of a function f(x) as x approaches a is the value that the function approaches as x gets arbitrarily large.

Answer

The limit of a function f(x) as x approaches a does not exist if the function is not continuous at a.

Question 25

According to the definition of a limit, what does it mean for the limit of f(x) as x approaches a to be L?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 26

Which of the following functions has a limit at a point but is not continuous at that point?

Accepted Answer

f(x) = |x| at x = 0

Answer

f(x) = x^2 at x = 0

Answer

f(x) = sin(x) at x = 0

Answer

f(x) = e^x at x = 0

Question 27

State the squeeze theorem.

Accepted Answer

If f(x) ≤ g(x) ≤ h(x) for all x in an interval containing a, and if limx→a f(x) = limx→a h(x) = L, then limx→a g(x) = L.

Answer

If f(x) ≤ g(x) ≤ h(x) for all x in an interval containing a, then limx→a f(x) = limx→a h(x) = L.

Answer

If f(x) ≤ g(x) ≤ h(x) for all x in an interval containing a, then limx→a g(x) = L.

Answer

If limx→a f(x) = limx→a h(x) = L, then f(x) ≤ g(x) ≤ h(x) for all x in an interval containing a.

Question 28

Which of the following statements about the limit of a function is true?

Accepted Answer

The limit of a function may not exist.

Answer

The limit of a function is always equal to the value of the function at the limit point.

Answer

The limit of a function is always zero.

Answer

The limit of a function is always unique.

Question 29

Which of the following is an example of a one-sided limit?

Accepted Answer

The limit of f(x) as x approaches a from the right (x > a)

Answer

The limit of f(x) as x approaches infinity

Answer

The limit of f(x) as x approaches negative infinity

Answer

The limit of f(x) as x approaches a

Question 30

Which of the following accurately defines the limit of a function, f(x)?

Accepted Answer

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

Answer

The slope of the tangent line to the graph of f(x) at x = a.

Answer

The value that f(x) approaches as x gets arbitrarily close to a.

Question 31

Which property of limits is demonstrated by the following equation: lim_(x->a) [f(x) + g(x)] = lim_(x->a) f(x) + lim_(x->a) g(x)?

Accepted Answer

The sum of the limits of two functions is equal to the limit of the sum of the functions.

Answer

The limit of a constant times a function is equal to the constant times the limit of the function.

Answer

The limit of the reciprocal of a function is equal to the reciprocal of the limit of the function.

Question 32

Evaluate the limit of the piecewise-defined function h(x) = { x^2, if x ≥ 0; -x^2, if x < 0 } as x approaches 0.

Accepted Answer

0

Answer

-1

Answer

1

Answer

The limit does not exist.

Question 33

Which of the following is NOT a valid theorem for evaluating limits?

Accepted Answer

The Product Rule for Limits

Answer

L'Hôpital's Rule

Answer

The Squeeze Theorem

Answer

The Sandwich Theorem

Question 34

What is the geometric interpretation of the limit of a function f(x) as x approaches a?

Accepted Answer

As x gets arbitrarily close to a, the distance between the graph of f(x) and the horizontal line y = L approaches 0.

Answer

The slope of the graph of f(x) at x = a is L.

Question 35

Which of the following functions satisfies the condition: lim_(x->∞) k(x) = 2?

Accepted Answer

k(x) = 2 + 1/x

Answer

k(x) = 1/x

Answer

k(x) = x^2

Question 36

How does a limit differ from an exact value in the context of functions?

Accepted Answer

A limit describes the behavior of a function as the input approaches a specific value, while an exact value is a specific number that the function takes at a specific input.

Answer

A limit is always an exact value, but an exact value is not always a limit.

Question 37

In the context of limits, what does the term 'ε-δ' represent?

Accepted Answer

The formal definition of a limit

Answer

The derivative of a function

Answer

The equation of a tangent line

Answer

The integral of a function

Question 38

In calculus, what does the concept of a limit describe?

Accepted Answer

The value a function approaches as its input gets closer to a specific value.

Answer

The maximum or minimum value a function can attain.

Answer

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

Question 39

When we say that a function f(x) has a limit of L as x approaches a, what does this mean?

Accepted Answer

As x gets infinitely close to a, the value of f(x) gets infinitely close to L.

Answer

f(x) is continuous at x = a.

Answer

f(a) is equal to L.

Question 40

Can a function have a limit as x approaches infinity even if the function doesn't have a defined value at infinity?

Accepted Answer

Yes, the limit can exist even if the function is undefined at infinity.

Answer

No, the limit must be equal to the value of the function at infinity.

Answer

The limit always exists if the function is continuous.

Question 41

If the limit of f(x) as x approaches 'a' is L, and the limit of g(x) as x approaches 'a' is M, what is the limit of the sum of f(x) and g(x) as x approaches 'a'?

Accepted Answer

L + M

Answer

L - M

Answer

L * M

Question 42

What is the limit of the function f(x) = 1/x as x approaches 0 from the right (i.e., x gets closer to 0 but remains positive)?

Accepted Answer

Positive infinity

Answer

0

Answer

Negative infinity

Answer

Does not exist

Question 43

The statement 'lim_(x->a) f(x) = L' means that:

Accepted Answer

We can make the value of f(x) as close to L as we want by making x sufficiently close to a.

Answer

f(x) is continuous at x = a

Answer

f(a) = L

Question 44

If the limit of f(x) as x approaches 'a' is L, and the limit of g(x) as x approaches 'a' is M, what is the limit of the product of f(x) and g(x) as x approaches 'a'?

Accepted Answer

L * M

Answer

L + M

Answer

L - M

Question 45

What does it mean to say that the limit of a function f(x) as x approaches a is L?

Accepted Answer

As x gets arbitrarily close to a, f(x) gets arbitrarily close to L.

Answer

f(x) is equal to L for all values of x near a.

Answer

f(a) is equal to L.

Question 46

Which of these statements is NOT true about the limit of a function?

Accepted Answer

The limit of a function always exists.

Answer

The limit of a function can be undefined.

Answer

The limit of a function can be a finite number.

Answer

The limit of a function can be positive or negative infinity.

Question 47

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

Accepted Answer

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

Answer

f(a) = L

Answer

f(x) is continuous at x = a

Question 48

Which of the following functions has a limit as x approaches 0?

Accepted Answer

f(x) = sin(x)/x

Answer

f(x) = 1/x

Answer

f(x) = |x|/x

Question 49

If the limit of f(x) as x approaches a exists and the limit of g(x) as x approaches a exists, what can we conclude about the limit of f(x) + g(x) as x approaches a?

Accepted Answer

It exists and is equal to the sum of the individual limits.

Answer

It does not necessarily exist.

Answer

It exists, but may not be equal to the sum of the individual limits.

Question 50

What is the limit of (1/x) as x approaches infinity?

Accepted Answer

0

Answer

Infinity

Answer

1

Answer

Does not exist

Question 51

Given the function f(x) = {x^2 if x < 2, 4 if x = 2, x + 2 if x > 2}, what is the limit of f(x) as x approaches 2?

Accepted Answer

4

Answer

6

Answer

2