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ch01_06

# ch01_06 - 1-49 SECTION 1.6 Formal Definition of the Limit...

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1-49 SECTION 1.6 . . Formal Definition of the Limit 121 What is the significance of the two vertical asymptotes? Ex- plain in physical terms what type of shot corresponds to each vertical asymptote. Estimate the minimum value of v 0 (call it v min ). Explain why it is easier to shoot a ball with a small ini- tial velocity. There is another advantage to this initial velocity. Assume that the basket is 2 ft in diameter and the ball is 1 ft in diameter. For a free throw, L = 15 ft is perfect. What is the maximum horizontal distance the ball could travel and still go in the basket (without bouncing off the backboard)? What is the minimum horizontal distance? Call these numbers L max and L min . Find the angle θ 1 corresponding to v min and L min and the angle θ 2 corresponding to v min and L max . The difference | θ 2 θ 1 | is the angular margin of error. Brancazio has shown that the angular margin of error for v min is larger than for any other initial velocity. 2. In applications, it is common to compute lim x →∞ f ( x ) to de- termine the “stability” of the function f ( x ). Consider the function f ( x ) = xe x . As x → ∞ , the first factor in f ( x ) goes to , but the second factor goes to 0. What does the product do when one term is getting smaller and the other term is getting larger? It depends on which one is chang- ing faster. What we want to know is which term “domi- nates.” Use graphical and numerical evidence to conjecture the value of lim x →∞ ( xe x ). Which term dominates? In the limit lim x →∞ ( x 2 e x ), which term dominates? Also, try lim x →∞ ( x 5 e x ). Based on your investigation, is it always true that exponentials dominate polynomials? Are you positive? Try to determine which type of function, polynomials or logarithms, domi- nates. 1.6 FORMAL DEFINITION OF THE LIMIT We have now spent many pages discussing various aspects of the computation of limits. This may seem a bit odd, when you realize that we have never actually defined what a limit is. Oh, sure, we have given you an idea of what a limit is, but that’s about all. Once again, we have said that lim x a f ( x ) = L , if f ( x ) gets closer and closer to L as x gets closer and closer to a . So far, we have been quite happy with this somewhat vague, although intuitive, de- scription. In this section, however, we will make this more precise, and you will begin to see how mathematical analysis (that branch of mathematics of which the calculus is the most elementary study) works. Studying more advanced mathematics without an understanding of the precise defi- nition of limit is somewhat akin to studying brain surgery without bothering with all that background work in chemistry and biology. In medicine, it has only been through a careful examination of the microscopic world that a deeper understanding of our own macroscopic world has developed, and good surgeons need to understand what they are doing and why they are doing it. Likewise, in mathematical analysis, it is through an understanding of the

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ch01_06 - 1-49 SECTION 1.6 Formal Definition of the Limit...

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