Chapter 3

Limits and the Derivative

Section 2

Infinite Limits and Limits at Infinity

(Part 1)

2Barnett/Ziegler/Byleen Business Calculus 12e

Objectives for Section 3.2 Infinite Limits and Limits at Infinity

The student will understand the concept of infinite limits. The student will be able to calculate limits at infinity.

3Barnett/Ziegler/Byleen Business Calculus 12e

Example 1

Recall from the first lesson:

lim𝑥→ 0−

1𝑥

=¿ lim𝑥→ 0+¿ 1

𝑥=¿¿

¿ lim𝑥→ 0

1𝑥

=¿− ∞ ∞ 𝐷𝑁𝐸

4Barnett/Ziegler/Byleen Business Calculus 12e

Infinite Limits and Vertical Asymptotes

Definition:

If the graph of y = f (x) has a vertical asymptote of x = a, then as x approaches a from the left or right, then f(x) approaches either or -.

Vertical asymptotes (and holes) are called points of discontinuity.

5Barnett/Ziegler/Byleen Business Calculus 12e

Example 2

Let

Identify all holes and asymptotes and find the left and right hand limits as x approaches the vertical asymptotes.

1

22

2

x

xxxf

𝑓 (𝑥 )=(𝑥+2)(𝑥−1)(𝑥+1)(𝑥− 1)

¿𝑥+2𝑥+1

𝐻𝑜𝑙𝑒 :(1 ,1.5)𝑉𝐴 :𝑥=−1

𝐻𝐴 : 𝑦=1

6Barnett/Ziegler/Byleen Business Calculus 12e

Example 2 (continued)

2

2

2( )

1

x xf x

x

Vertical Asymptote

Hole

lim𝑥→ −1−

𝑥+2𝑥+1

=¿¿lim

𝑥→ −1+¿ 𝑥+2𝑥+1

=¿ ¿¿

¿lim𝑥→ −1

𝑥+2𝑥+1

=¿¿− ∞ ∞ 𝐷𝑁𝐸

Horizontal Asymptote

7Barnett/Ziegler/Byleen Business Calculus 12e

Example 3

Let

Identify all holes and asymptotes and find the left and right hand limits as x approaches the vertical asymptotes.

𝑓 ( x )= 1

(𝑥− 2)2 𝑁𝑜𝐻𝑜𝑙𝑒𝑠

𝑉𝐴 :𝑥=2

𝑓 (𝑥 )= 1

(𝑥−2)2

𝐻𝐴 : 𝑦=0

8

Example 3 (continued)

Barnett/Ziegler/Byleen Business Calculus 12e

lim𝑥→ 2−

1

(𝑥−2)2 =¿ ¿∞ ∞lim

𝑥→ 2+¿ 1

(𝑥− 2)2 =¿¿ ¿

¿∞lim

𝑥→ 2

1

(𝑥− 2)2 =¿¿

9Barnett/Ziegler/Byleen Business Calculus 12e

Limits at Infinity

• We will now study limits as x ±.

• This is the same concept as the end behavior of a graph.

10

End Behavior Review

Barnett/Ziegler/Byleen Business Calculus 12e

Odd degreePositiveleading

coefficient

Odd degreeNegativeleading

coefficient

Even degreePositiveleading

coefficient

Even degreeNegativeleading

coefficient

11

Polynomial Functions

Ex 4: Evaluate each limit.

Barnett/Ziegler/Byleen Business Calculus 12e

lim𝑥→ −∞

𝑥2=¿¿

lim𝑥→+∞

𝑥2=¿¿

lim𝑥→ −∞

−3 𝑥4=¿¿

lim𝑥→+∞

− 3𝑥4=¿¿

lim𝑥→ −∞

6 𝑥3=¿

lim𝑥→+∞

6 𝑥3=¿

lim𝑥→ −∞

−5 𝑥3=¿

lim𝑥→+∞

−5 𝑥3=¿

∞

∞

− ∞

− ∞

− ∞

∞

∞

− ∞

12

Rational Functions

If a rational function has a horizontal asymptote, then it determines the end behavior of the graph.

If f(x) is a rational function, then

Barnett/Ziegler/Byleen Business Calculus 12e

lim𝑥→ ± ∞

𝑓 (𝑥 )=h𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒𝑣𝑎𝑙𝑢𝑒

13

Rational Functions

Ex 5: Evaluate

Barnett/Ziegler/Byleen Business Calculus 12e

𝑓 (𝑥 )= 5𝑥+2 𝐻𝐴 : 𝑦=0

Because the degree of the numerator < degree of the

denominator.

lim𝑥→ ∞

𝑓 (𝑥 )=¿¿

lim𝑥→ −∞

𝑓 (𝑥 )=¿¿

0

0

lim𝑥→ ± ∞

𝑓 (𝑥 )

14

Rational Functions

Ex 6: Evaluate

Barnett/Ziegler/Byleen Business Calculus 12e

𝐻𝐴 : 𝑦=32

Because the degree of the numerator = degree of the

denominator.

lim𝑥→ ∞

𝑓 (𝑥 )=¿¿

lim𝑥→ −∞

𝑓 (𝑥 )=¿¿

32

32

2

2

3 5 9

2 7

x xy

x

lim𝑥→ ± ∞

𝑓 (𝑥 )

15

Rational Functions

If a rational function doesn’t have a horizontal asymptote, then to determine its end behavior, take the limit of the ratio of the leading terms of the top and bottom.

Barnett/Ziegler/Byleen Business Calculus 12e

16

Rational Functions

Ex 7: Evaluate

Barnett/Ziegler/Byleen Business Calculus 12e

𝐻𝐴 :𝑁𝑜𝑛𝑒Because the degree of the numerator > degree of the

denominator.

lim𝑥→ ∞

2 𝑥5

6 𝑥3 =¿¿

𝑓 (𝑥 )= 2 𝑥5 −𝑥3− 16 𝑥3+2 𝑥2−7

lim𝑥→ ∞

𝑥2

3=¿¿∞

lim𝑥→ −∞

2𝑥5

6 𝑥3 =¿ ¿lim𝑥→ −∞

𝑥2

3=¿¿∞

lim𝑥→ ± ∞

𝑓 (𝑥 )

17

Rational Functions

Ex 8: Evalaute

Barnett/Ziegler/Byleen Business Calculus 12e

𝐻𝐴 :𝑁𝑜𝑛𝑒

lim𝑥→ ∞

5 𝑥6

2 𝑥5 =¿¿

𝑓 (𝑥 )= 5𝑥6+3 𝑥2 𝑥5 −𝑥− 5

lim𝑥→ ∞

5 𝑥2

=¿¿∞

lim𝑥→ −∞

5 𝑥6

2𝑥5 =¿¿lim𝑥→ −∞

5 𝑥2

=¿¿− ∞

lim𝑥→ ± ∞

𝑓 (𝑥 )

18

Homework

Barnett/Ziegler/Byleen Business Calculus 12e

#3-2A: Pg 150

(3-15 mult. of 3,17, 19, 31-35 odd, 39, 43, 45, 61, 65)

Chapter 3

Limits and the Derivative

Section 2

Infinite Limits and Limits at Infinity

(Part 2)

20Barnett/Ziegler/Byleen Business Calculus 12e

Objectives for Section 3.2 Infinite Limits and Limits at Infinity

The student will be able to solve applications involving limits.

2 ∞ & >

21

Application: Business

T & C surf company makes surfboards with fixed costs at $300 per day. One day, they made 20 boards and total costs were $5100.

A. Assuming the total cost per day is linearly related to the number of boards made per day, write an equation for the cost function.

B. Write the equation for the average cost function.

C. Graph the average cost function:

D. What does the average cost per board approach as production increases?

Barnett/Ziegler/Byleen Business Calculus 12e

22

Application: Business

T & C surf company makes surfboards with fixed costs at $300 per day. One day, they made 20 boards and total costs were $5100. Assuming the total cost per day is linearly related to the

number of boards made per day, write an equation for the cost function.

Barnett/Ziegler/Byleen Business Calculus 12e

𝑦=𝑚𝑥+𝑏5100=𝑚(20)+300𝑚=240

h𝑇 𝑒𝑐𝑜𝑠𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑖𝑠 :𝐶 (𝑥 )=240 𝑥+300

23

Application: Business

T & C surf company makes surfboards with fixed costs at $300 per day. One day, they made 20 boards and total costs were $5100. Write the equation for the average cost function.

Barnett/Ziegler/Byleen Business Calculus 12e

𝐶 (𝑥 )=240𝑥+300𝑥

𝐶 (𝑥 )=𝐶 (𝑥)𝑥

24

Application: Business

T & C surf company makes surfboards with fixed costs at $300 per day. One day, they made 20 boards and total costs were $5100. Graph the average cost function:

Barnett/Ziegler/Byleen Business Calculus 12e

Number of surfboards

Average cost per

day

25

Application: Business

T & C surf company makes surfboards with fixed costs at $300 per day. One day, they made 20 boards and total costs were $5100. What does the average cost per board approach as

production increases?

Barnett/Ziegler/Byleen Business Calculus 12e

𝐶 (𝑥 )=240𝑥+300𝑥

As the number of boards increases, the average cost approaches $240 per board.

Number of surfboards

Average cost per

day

26

Application: Medicine

A drug is administered to a patient through an IV drip. The drug concentration (mg per milliliter) in the patient’s bloodstream t hours after the drip was started is modeled by the equation:

A. What is the drug concentration after 2 hours?

B. Evaluate and interpret the meaning of the limit:

Barnett/Ziegler/Byleen Business Calculus 12e

𝐶 (𝑡 )=5 𝑡 (𝑡+50 )𝑡 3+100

lim𝑡→ ∞

𝐶 (𝑡)

27

Application: Medicine

A drug is administered to a patient through an IV drip. The drug concentration (mg per milliliter) in the patient’s bloodstream t hours after the drip was started is modeled by the equation:

What is the drug concentration after 2 hours?

Barnett/Ziegler/Byleen Business Calculus 12e

𝐶 (𝑡 )=5 𝑡 (𝑡+50 )𝑡 3+100

𝐶 (2 )=5 (2)(2+50 )

23+100≈ 4.8

After 2 hours, the concentration of the drug is 4.8 mg/ml.

28

Application: Medicine

A drug is administered to a patient through an IV drip. The drug concentration (mg per milliliter) in the patient’s bloodstream t hours after the drip was started is modeled by the equation:

Evaluate and interpret the meaning of the limit:

Barnett/Ziegler/Byleen Business Calculus 12e

𝐶 (𝑡 )=5 𝑡 (𝑡+50 )𝑡 3+100

lim𝑡→ ∞

𝐶 (𝑡)

lim𝑡→ ∞

5 𝑡 (𝑡+50 )𝑡 3+100

=0

As time passes, the drug concentration approaches 0 mg/ml.

29

Homework

Barnett/Ziegler/Byleen Business Calculus 12e

#3-2B: Pg 150(2-8 even, 11, 13, 18,

34, 36, 37, 41, 49, 63, 67, 69, 73, 76)