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Formal Definition of Limit Problem: In Chapter 2, you will learn that the formal

Calculus: Concepts and Applications | 2nd Edition | ISBN: 9781559536547 | Authors: Paul A. Foerster ISBN: 9781559536547 285

Solution for problem c3 Chapter 1-6

Calculus: Concepts and Applications | 2nd Edition

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Calculus: Concepts and Applications | 2nd Edition | ISBN: 9781559536547 | Authors: Paul A. Foerster

Calculus: Concepts and Applications | 2nd Edition

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Problem c3

Formal Definition of Limit Problem: In Chapter 2, you will learn that the formal definition of limit is L = limxc f(x) if and only if for any positive number epsilon, no matter how small there is a positive number delta such that if x is within delta units of c, but not equal to c, then f(x) is within epsilon units of L. Notes: limxc f(x) is read the limit of f(x) as x approaches c. Epsilon is the Greek lowercase letter . Delta is the Greek lowercase letter . 123456 1 2 3 4 5 6 x (s) f(x) (ft) Figure 1-6d Figure 1-6d shows the graph of the averagevelocity in ft/s for a moving object from 3 s tox s given by the functionf(x) = 4x2 19x + 21x 3From the graph you can see that 5 is the limitof f(x) as x approaches 3 (the instantaneousvelocity at x = 3), but that r (3) is undefinedbecause of division by zero.a. Show that the (x 3) in the denominatorcan be canceled out by first factoring thenumerator, and that 5 is the value of thesimplified expression when x = 3.b. If = 0.8 unit, on a copy of Figure 1-6d showthe range of permissible values of f(x) andthe corresponding interval of x-values thatwill keep f(x) within 0.8 unit of 5.c. Calculate the value of to the right of 3 inpart b by substituting 3 + for x and 5.8 forf(x), then solving for . Show that you getthe same value of to the left of 3 bysubstituting 3 for x and 4.2 for f(x).d. Suppose you must keep f(x) within unitsof 5, but you havent been told the value of. Substitute 3 + for x and 5 + for f(x).Solve for in terms of . Is it true that thereis a positive value of for each positivevalue of , no matter how small, as requiredby the definition of limit?e. In this problem, what are the values of L andc in the definition of limit? What is thereason for the clause . . . but not equal to cin the definition?

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Chapter 1-6, Problem c3 is Solved
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Textbook: Calculus: Concepts and Applications
Edition: 2
Author: Paul A. Foerster
ISBN: 9781559536547

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Formal Definition of Limit Problem: In Chapter 2, you will learn that the formal

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