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Unformatted text preview: 1-49SECTION 1.6..Formal Definition of the Limit121What is the significance of the two vertical asymptotes? Ex-plain in physical terms what type of shot corresponds to eachvertical asymptote. Estimate the minimum value ofv(call itvmin). 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 1ft in diameter. For a free throw,L=15 ft is perfect. What isthe maximum horizontal distance the ball could travel and stillgo in the basket (without bouncing off the backboard)? Whatis the minimum horizontal distance? Call these numbersLmaxandLmin. Find the angle1corresponding tovminandLminandthe angle2corresponding tovminandLmax. The difference|21|is the angular margin of error. Brancazio has shownthat the angular margin of error forvminis larger than for anyother initial velocity.2.In applications, it is common to compute limxf(x) to de-termine the stability of the functionf(x). Consider thefunctionf(x)=xex. Asx , the first factor inf(x)goes to, but the second factor goes to 0. What does theproduct do when one term is getting smaller and the otherterm 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 conjecturethe value of limx(xex). Which term dominates? In the limitlimx(x2ex), which term dominates? Also, try limx(x5ex).Based on your investigation, is it always true that exponentialsdominate polynomials? Are you positive? Try to determinewhich type of function, polynomials or logarithms, domi-nates.1.6FORMAL DEFINITION OF THE LIMITWe 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 actuallydefinedwhat a limitis. Oh, sure, we have given you anideaof what a limit is, but thats about all. Once again,we have said thatlimxaf(x)=L,iff(x) gets closer and closer toLasxgets closer and closer toa.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 tosee howmathematical analysis(that branch of mathematics of which the calculus is themost 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 thatbackground work in chemistry and biology. In medicine, it has only been through a carefulexamination of the microscopic world that a deeper understanding of our own macroscopicworld has developed, and good surgeons need to understand what they are doingand whythey are doing it. Likewise, in mathematical analysis, it is through an understanding of themicroscopic behavior of functions (such as the precise definition of limit) that a deeperunderstanding of the mathematics will come about.understanding of the mathematics will come about....
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This note was uploaded on 10/03/2010 for the course ENG 42325 taught by Professor Lagerstrom during the Spring '10 term at UC Davis.
- Spring '10