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 | 1. Suppose your friend says, “You can think of the limit of f (x) as x approaches a as what f(a) should be. ” Critique this statement. What does it mean? Does it provide important insight? Is there anything misleading about it? Replace the phrase in italics with your own best description of what the limit is. |
| | 2. Your friend says, “The limit is a prediction of what f (a) will be. ” Compare and contrast this statement to the one in exercise 1. Does the inclusion of the word prediction make the limit idea seem more useful and important? |
| | 3. We have emphasized that  does not depend on the actual value of f (a) , or even on whether f (a) exists. In principle, functions such as  are as “normal” as functions such as g(x) = x2. With this in mind, explain why it is important that the limit concept is independent of how (or whether) f (a) is defined. |
| | 4. The most common limit encountered in everyday life is the speed limit. Describe how this type of limit is very different from the limits discussed in this section. |
| 5. For the function graphed below, identify each limit or state that it does not exist. |
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(a)  | (b)  |
(c)  | (d)  |
(e)  | (f)  |
(g)  | (h)  |
(i)  | (j)  |
| 
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| 6. For the function graphed below, identify each limit or state that it does not exist. |
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| (a) |  | (b)  |
| (c) |  | (d)  |
| (e) |  | (f)  |
| (g) |  | (h)  |
| (i) |  | (j)  |
| 
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| 7. Evaluate f (1.5), f (1.1) , f (1.01) and f (1.001) and conjecture a value for  for  . Evaluate f (0.5) , f (0.9) , f (0.99) and f (0.999) and conjecture a value for  for  . Does  exist? |
| 8. Evaluate f ( -1.5) , f ( -1.1) , f ( -1.01) and f ( -1.001) and conjecture a value for  for  . Evaluate f ( -0.5), f ( -0.9) , f ( -0.99) and f ( -0.999) and conjecture a value for  for  . Does exist? |
| In exercises 9-16, use numerical and graphical evidence to conjecture values for each limit.
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| 9.  | 10.  |
| 11.  | 12. 
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| 13.  | 14.  |
| 15.  | 16. 
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| In exercises 17-26, use numerical and graphical evidence to conjecture whether or not the limit at x = a exists. If not, describe what is happening at x = a graphically.
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| 17.  | 18.  |
| 19.  | 20.  |
| 21.  | 22.  |
| 23.  | 24.  |
| 25.  | 26. 
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| 27. Use your results from exercises 21 and 22 to investigate the following. Suppose that f (x) and g(x) are polynomials with g(a) = 0 and f (a) 0. What can you conjecture about
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| 28. Use your results from exercises 19 and 20 to investigate the following. Suppose that f (x) and g(x) are polynomials with f (a) = 0 and g(a) 0. What can you conjecture about  ? |
| In exercises 29-32, sketch a graph of a function with the given properties.
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| 29. f ( -1) = 2 , f (0) = - 1 , f (1) = 3  does not exist. |
| 30. f (x) = 1 for - 2 x 1 ,  and  =. |
| 31.  and  . |
| 32.  , f (2) = 3 and  does not exist. |
| 33. Consider the following arguments concerning  . First, as x > 0 approaches 0,  increases without bound; since sin t oscillates for increasing t, the limit does not exist. Second: taking x = 1, 0.1, 0.01 and so on, we compute sin  = sin 10 = sin 100 =  = 0; therefore the limit equals 0.
Which argument sounds better to you? Explain. Explore the limit and determine which answer is correct. |
| 34. Consider the following argument concerning  As x approaches 0,  increases without bound and  decreases
without bound. Since et approaches 0 as t decreases without bound, the limit equals 0. Discuss all the errors made in this argument. |
| 35. Numerically estimate the limits  and
 Note that the function values for x > 0
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| increase as x decreases, while for x < 0 the function values decrease as x increases. Explain why this indicates that the limit  is between function values for positive and |
| negative x's. Approximate this limit correct to eight digits. |
| 36. Explain what is wrong with the following logic (you may use exercise 35 to convince yourself that the answer is wrong, but discuss the logic without referring to exercise 35): as x 0 , it is clear that (1 + x) 1. Since 1 raised to any power is 1,  . |
| 37. Numerically estimate  . Try to numerically estimate |
|  . If your computer has difficulty evaluating the
function for negative x's, explain why. |
| 38. Explain what is wrong with the following logic (note from exercise 37 that the answer is accidentally correct): since 0 to any power is 0,  . |
| 39. Give an example of a function f (x) such that  exists but f (0) does not exist. Give an example of a function g(x) such that g(0) exists but  does not exist. |
| 40. Give an example of a function f (x) such that  exists and f (0) exists, but  . |
| 41. In the text, we described  as meaning “as x gets closer and closer to a, f (x) is getting closer and closer to L. ” As x gets closer and closer to 0, it is true that x2 gets closer and closer to -0.01 , but it is certainly not true that  . Try to modify the description of limit to make |
| it clear that  . We will explore a very precise |
| definition of limit in Section 1.5. |
| 42. In
Figure 1.8, the final position of the knuckleball at time t = 0.68 is shown as a function of the rotation rate  . The batter must decide at timet = 0.4 whether or not to swing at the pitch. At t = 0.4, the left/right position of the ball is given by h() =  and compare to
Figure 1.8. |
| Conjecture the limit of h() as 0. For = 0, is there any difference in ball position between what the batter sees at t = 0.4 and what he tries to hit at t = 0.68?
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 | 43. In a situation similar to that of example 1.6, the left/right position of a knuckleball pitch in baseball can be modeled by  , where t is time measured in seconds ( 0 t 0.68 ) and is the rotation rate of the ball measured in radians per second. In
Example 1.6, we chose a specific t - value and evaluated the limit as 0. While this gives us some information about which rotation rates produce hard-to-hit pitches, a clearer picture emerges if we look at f (t ) over its entire domain. Set = 10 and graph the resulting function  for 0 t 0.68. Imagine looking at a pitcher from above and try to visualize a baseball starting at the pitcher's hand at t = 0 and finally reaching the batter, at t = 0.68. Repeat this with = 5 , = 1 , = 0.1 and whatever values of you think would be interesting. Which values of produce hard-to-hit pitches? |
| | 44. In this exercise, the results you get will depend on the accuracy of your computer or calculator. Work this exercise and compare your results with your classmates' results. We will investigate  . Start by verifying the calculations presented in the table: |
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| x | f (x) |
| 0.1 | - 0.499583... |
| 0.01 | - 0.49999583... |
| 0.001 | - 0.4999999583... |
| Describe as precisely as possible the pattern shown here. What would you predict for f (0.0001) ? f (0.00001) ? Does your computer or calculator give you this answer? If you continue trying powers of 0.1 (0.000001, 0.0000001, etc.) you should eventually be given a displayed result of - 0.5. Do you think this is exactly correct, or has the answer just been rounded off? Why is rounding off inescapable? It turns out that - 0.5 is the exact value for the limit, so the round-off here is somewhat helpful. However, if you keep evaluating the function at smaller and smaller values of x, you will eventually see a reported function value of 0. This round-off error is not so benign; we discuss this error in section 1.6. For now, evaluate cos x at the current value of x and try to explain where the 0 came from. |
