Limits

Question 1

What is the limit as x approaches 2 for the function -x + 6?

Answer 1

4

This is calculated by substituting x with 2 in the function.

Question 2

When substituting values in a limit function, what happens if the equation is a fraction?

Answer 2

It is not as simple as plugging in values directly because you would get 0/0

You may need to analyze the behavior from both sides of the limit.

Question 3

What is the limit as x approaches 2 for the fraction (x-4)/(x-2)?

Answer 3

4

Approaching from the left and right sides yields values that converge to 4.

Question 4

Fill in the blank: As x approaches 2, the limit of (2.1-4)/(2.1-2) approaches _______.

Answer 4

4.1

Question 5

Fill in the blank: As x approaches 2, the limit of (2.01-4)/(2.01-2) approaches _______.

Answer 5

4.01

Question 6

Fill in the blank: As x approaches 2, the limit of (1.93-4)/(1.93-2) approaches _______.

Answer 6

3.9

Question 7

What does it mean when a limit converges into a certain value?

Answer 7

It approaches a specific value as the input gets closer to a point

In this case, the limit approaches 4 as x approaches 2.

Question 8

What is one method to find the limit of a function if direct substitution is not possible?

Answer 8

Factorising the function

This can help simplify the expression to evaluate the limit.

Question 9

What is the limit as x approaches 2 of (x³ - 4)/(x - 2)?

Answer 9

2 + 2 = 4

This limit can be evaluated by factorizing the numerator.

Question 10

In the expression (3 - x)(x - 3), what is the relationship between (3 - x) and (x - 3)?

Answer 10

(3 - x) = -1(x - 3)

This shows that they are negatives of each other.

Question 11

True or False: You can directly substitute x = 3 into (x - 3)/(x - 3) without simplification.

Answer 11

False

The expression is indeterminate at x = 3 and requires simplification.

Question 12

Fill in the blank: The limit as x approaches 3 of (3 - x)/(x - 3) equals _______.

Answer 12

-1

Question 13

What is the first step to solve a problem involving a limit with a square root?

Answer 13

Multiply the top and the conjugate of the square root

Question 14

What is the limit notation for approaching from the left?

Answer 14

lim x→a-

Indicates the limit as x approaches a from the left side.

Question 15

What is the limit notation for approaching from the right?

Answer 15

lim x→a+

Indicates the limit as x approaches a from the right side.

Question 16

What does it mean if lim x→a- and lim x→a+ are equal?

Answer 16

lim x→a exists

The limit exists if both one-sided limits match.

Question 17

What does it mean if lim x→a- and lim x→a+ are different?

Answer 17

Limit does not exist (DNE)

If the left-hand limit and right-hand limit do not match, the limit is undefined.

Question 18

How is f(a) determined when discussing limits?

Answer 18

Closed dot’s y-value

The value of the function at x=a is determined by the closed dot on the graph.

Question 19

What happens if the x-value is near a vertical asymptote?

Answer 19

Check limits before and after

The behavior of the function around vertical asymptotes can affect the limit.

Question 20

What is the result of lim x→1 if it approaches from the left and right yield different values?

Answer 20

DNE

This indicates the limit does not exist when the left and right limits are unequal.

Question 21

Fill in the blank: To find the limit as x approaches a, if the previous values match, write down that value. If they are different, the value does not exist (____).

Answer 21

DNE

Question 22

True or False: The limit can exist even if the function is not defined at that point.

Answer 22

True

Limits can exist independently of the function’s value at that point.

Question 23

What is a vertical asymptote?

Answer 23

A vertical asymptote occurs when the function approaches infinity or negative infinity as the x-value approaches a certain point

It indicates that the function is undefined at that point.

Question 24

What does lim f(x) as x approaches 1 from the left equal if the vertical asymptote is going down?

Answer 24

-∞

This indicates that the function decreases without bound as it approaches the asymptote.

Question 25

What does lim f(x) as x approaches 1 from the right equal if the vertical asymptote is going up?

Answer 25

+∞

This indicates that the function increases without bound as it approaches the asymptote.

Question 26

What is a limit defined as continuous?

Answer 26

1. f(a) is defined
2. lim f(x) exists as x approaches a
3. Lim f(x) = f(a) as x approaches a

These conditions ensure the function behaves consistently at the point of interest.

Question 27

What is the value of lim x→0 of x?

Answer 27

+∞

This indicates that as x approaches 0 from the right, the function grows without bound.

Question 28

What is the value of lim x→0 of x from the left?

Answer 28

-∞

This shows that as x approaches 0 from the left, the function decreases without bound.

Question 29

What does lim x→+∞ of 1+x equal?

Answer 29

+∞

This indicates that as x grows larger, the value of the function continues to increase without bound.

Question 30

What does it mean when a function is bottom heavy?

Answer 30

It equals O

Bottom heavy functions approach zero as they grow.

Question 31

What does it mean when a function is top heavy?

Answer 31

It equals +∞

Top heavy functions diverge to infinity.

Question 32

What theorem is used to find the limit of a function?

Answer 32

Squeeze theorem

The Squeeze theorem helps determine limits by comparison.

Question 33

What is the form of the Squeeze theorem?

Answer 33

If lim h(x) < lim f(x) < lim g(x), then lim f(x) = L

This applies as x approaches a specific value.

Question 34

In the Squeeze theorem, if lim h(x) = L and lim g(x) = L, what can be concluded about lim f(x)?

Answer 34

lim f(x) = L

This establishes that f(x) is squeezed to L.

Question 35

What is the limit of 8 - x³ as x approaches 0?

Answer 35

8

Evaluating the limit gives 8 - 0³ = 8.

Question 36

What is the limit of 8 + x³ as x approaches 0?

Answer 36

8

Evaluating the limit gives 8 + 0³ = 8.

Question 37

What conclusion can be drawn from lim 8 - x³ < lim f(x) < lim 8 + x³?

Answer 37

lim f(x) = 8

This demonstrates that f(x) converges to 8 as x approaches 0.