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Chapter 3 Limits and the Derivative Section 2 Infinite Limits and Limits at Infinity (Part 1)

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1 Chapter 3 Limits and the Derivative Section 2 Infinite Limits and Limits at Infinity (Part 1)

2 2 Barnett/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.

3 3 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 Recall from the first lesson:

4 4 Barnett/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.

5 5 Barnett/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.

6 6 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 (continued) Vertical Asymptote Hole Horizontal Asymptote

7 7 Barnett/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.

8 8 Example 3 (continued) Barnett/Ziegler/Byleen Business Calculus 12e

9 9 Limits at Infinity We will now study limits as x  ± . This is the same concept as the end behavior of a graph.

10 10 End Behavior Review Barnett/Ziegler/Byleen Business Calculus 12e Odd degree Positive leading coefficient Odd degree Negative leading coefficient Even degree Positive leading coefficient Even degree Negative leading coefficient

11 11 Polynomial Functions  Ex 4: Evaluate each limit. Barnett/Ziegler/Byleen Business Calculus 12e

12 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

13 13 Rational Functions  Ex 5: Evaluate Barnett/Ziegler/Byleen Business Calculus 12e Because the degree of the numerator < degree of the denominator.

14 14 Rational Functions  Ex 6: Evaluate Barnett/Ziegler/Byleen Business Calculus 12e Because the degree of the numerator = degree of the denominator.

15 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 16 Rational Functions  Ex 7: Evaluate Barnett/Ziegler/Byleen Business Calculus 12e Because the degree of the numerator > degree of the denominator.

17 17 Rational Functions  Ex 8: Evalaute Barnett/Ziegler/Byleen Business Calculus 12e

18 18 Homework Barnett/Ziegler/Byleen Business Calculus 12e

19 Chapter 3 Limits and the Derivative Section 2 Infinite Limits and Limits at Infinity (Part 2)

20 20 Barnett/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.

21 21 Application: Business Barnett/Ziegler/Byleen Business Calculus 12e

22 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

23 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

24 24 Application: Business Barnett/Ziegler/Byleen Business Calculus 12e Number of surfboards Average cost per day

25 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 As the number of boards increases, the average cost approaches $240 per board. Number of surfboards Average cost per day

26 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

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

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

29 29 Homework Barnett/Ziegler/Byleen Business Calculus 12e


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