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

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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.

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

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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.

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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.

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

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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.

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

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

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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

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

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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

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

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

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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

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

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

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18 Homework Barnett/Ziegler/Byleen Business Calculus 12e

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

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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.

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

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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

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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

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

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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

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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

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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.

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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.

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29 Homework Barnett/Ziegler/Byleen Business Calculus 12e

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