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1.4 |
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 | 1. It may seem odd that we use  in describing limits but do not count as a real number. Discuss the existence of : is it a number or a concept? |
| | 2. In example 4.6, we dealt with the “indeterminate form”  . Thinking of a limit of as meaning “getting very large” and a limit of 0 as meaning “getting very close to 0” explain why the following are indeterminate forms:  ,  , - , and  0. Determine what the following nonindeterminate forms represent: + , - - , + 0 and 0/. |
| | 3. On your computer or calculator, graph y = 1/(x -2) and look for the horizontal asymptote y = 0 and the vertical asymptote x = 2. Most computers will draw a vertical line at x = 2 and will show the graph completely flattening out at y = 0 for large x's. Is this accurate? misleading? Most computers will compute the locations of points for adjacent x's and try to connect the points with a line. Why might this result in a vertical line at the location of a vertical asymptote? |
| | 4. Many students learn that asymptotes are lines which the graph gets closer and closer to without ever reaching. This is true for many asymptotes, but not all. Explain why vertical asymptotes are never reached or crossed. Explain why horizontal or slant asymptotes may, in fact, be crossed any number of times; draw one example. |
| In exercises 5-40, determine the following limits (answer as appropriate, with a number, , - or does not exist).
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| 5.  | 6.  |
| 7.  | 8.  |
| 9.  | 10.  |
| 11.  | 12.  |
| 13.  | 14.  |
| 15.  | 16.  |
| 17.  | 18. 
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| 19.  | 20.  |
| 21.  | 22.  |
| 23.  | 24.  |
| 25.  | 26.  |
| 27.  | 28.  |
| 29.  | 30.  |
| 31.  | 32.  |
| 33.  | 34.  |
| 35.  | 36.  |
| 37.  | 38.  |
| 39.  | 40. 
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| In exercises 41-50, determine all horizontal, slant and vertical asymptotes.
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| 41.  | 42.  |
| 43.  | 44.  |
| 45.  | 46.  |
| 47.  | 48.  |
| 49.  | 50. 
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| 51. Sketch the graph of f (x) = e - xcos x. Identify the horizontal asymptote. Is this asymptote approached in both directions (as x and x - )? How many times does the graph cross the horizontal asymptote? |
| 52. Explain why it is reasonable that  and  . |
| 53. In the exercises of section 1.1, we found that   . (It turns out that both limits equal the irrational number e.) Use this result and exercise 52 to argue that  . |
| 54. One of the reasons for saying that infinite limits do not exist is that we would otherwise invalidate Theorem 2.3 in section 1.2. Find examples of functions with infinite limits such that parts (ii) and (iv) of Theorem 2.3 are not correct. |
| 55. Suppose that the length of a small animal t days after birth is  What is the length of the animal at birth? What is the eventual length of the animal (i.e., the length as t )? |
| 56. Suppose that the length of a small animal t days after birth is  What is the length of the animal at birth? What is the eventual length of the animal (i.e., the length as t )? |
| 57. Suppose that the size of the pupil of a certain animal is given by f (x) (mm), where x is the intensity of the light on the pupil. If  , find the size of the pupil with no light and the size of the pupil with an infinite amount of light. |
| 58. Repeat exercise 57 with  . |
| 59. Modify the functions in exercises 57 and 58 to find a function f such that  and 
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| 60. After an injection, the concentration of drug in a muscle varies according to a function of time f (t ). Suppose that t is measured in hours and f (t ) = e -0.02t - e -0.42t . Find the limit of f (t ) both as t 0 and t and interpret both limits in terms of the concentration of drug. |
| 61. Suppose an object with initial velocity v0 = 0 ft/s and (constant) mass m slugs is accelerated by a constant force F pounds for t seconds. According to Newton's laws of motion, the object's speed will be vN = Ft/m. According to Einstein's theory of relativity, the object's speed will be vE =  , where c is the speed of light. Compute  and  . |
| 62. According to Einstein's theory of relativity, the mass of an object traveling at speed  is given by  , where c is the speed of light (about 9.8 108 ft/s). Compute  and explain why m0 is called the “rest mass. ” Compute  and discuss the implications. (What would happen if |
| you were traveling in a spaceship approaching the speed of light?) How much does the mass of a 192 - pound man ( m0 = 6 ) increase at the speed of 9000 ft/s (about 4 times the speed of sound)? |
| 63. Explain why  for any positive constant a. Although this is theoretically true, it is not necessarily useful in practice. The function e - atsin t is a simple model for a spring-mass system, such as the shock absorber on a car. Suppose t is measured in seconds and the car passengers cannot feel any vibrations less than 0.01 (inches). If shock absorber A has thevibration function e - t sin t and shock absorber B has the vibration function e - t /4sin t , determine graphically how long it will take before the vibrations damp out, that is,  0.01. Is the result  much consolation to the owner of shock absorber B? |
| 64. The speed of a 128 - lb sky-diver is given by (approximately)  ft/s, where k is a constant that depends on the orientation of the sky-diver's body. Compute  . Explain why this is called “terminal velocity. ” Compare the terminal velocity of sky-divers in the spread-eagle position ( k = 3 ) and the tuck position ( k = 1 ). How are sky-divers able to control their speed? |
| 65. State and prove a result analogous to Theorem 4.2 for  for n odd. |
| 66. State and prove a result analogous to Theorem 4.2 for  for n even. |
| 67. It is very difficult to find simple statements in calculus that are always true; this is one reason that a careful development of theory is so important. You may have heard the simple rule: to find the vertical asymptotes of  , simply set the denominator equal to 0 [i.e., solve h(x) = 0 ]. Give an example where h(a) = 0 but there is not a vertical asymptote at x = a. |
| 68. In exercise 67, you needed to find an example indicating that the following statement is not (necessarily) true: if h(a) = 0 , then  has a vertical asymptote at x = a. This is not true, but perhaps its converse is true: if  has a vertical asymptote at x = a , then h(a) = 0. Is this statement true? What if g and h are polynomials? |
| In exercises 69-74, use numerical evidence to conjecture a decimal representation for the limit. Check your answer with your computer algebra system (CAS); if your answers disagree, which one is correct?
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| 69.  | 70.  |
| 71.  | 72.  |
| 73.  | 74. 
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 | 75. Suppose you are shooting a basketball from a (horizontal) distance of L feet, releasing the ball from a location h feet below the basket. To get a perfect swish, it is necessary that the initial velocity v0 and initial release angle  0 satisfy the equation   . For a free throw, take L = 15 , h = 2 and g = 32 and graph v0 as a function of 0. What is the significance of the two vertical
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| asymptotes? Explain in physical terms what type of shot corresponds to each vertical asymptote. Estimate the minimum value of V0 (call it Vmin ). Explain why it is easier to shoot a ball with a small initial 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 (don't count bouncing off the backboard)? What is the minimum horizontal distance? Call these numbers  and  . Find the angle 1 corresponding to Vmin and Lmin and the angle 2 corresponding to Vmin and Lmax. The difference  is the angular margin of error. Brancazio has shown that the angular margin of error for Vmin is larger than for any other initial velocity. |
| | 76. In applications, it is common to compute  to determine 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 changing faster. What we want to know is which function “dominates. ” Use graphical and numerical evidence to conjecture the value of  . Which function dominates? In the limit  , which function dominates? Also, try  . 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, dominates. |
