 |
|
 | 1. Given your knowledge of the graphs of polynomials, explain why
Theorems 2.1, 2.2 and
2.4 are obvious. Name five non-polynomial functions for which limits can be evaluated by substitution. |
| | 2. Suppose that you can draw the graph of y = f (x) without lifting your pencil from your paper. Explain why  , for every value of a. |
| | 3. In one or two sentences, explain the Squeeze Theorem. Use a real-world analogy (e.g., having the functions represent the locations of three people as they walk) to indicate why it is true. |
| | 4. Given the graph in Figure 1.15 and the calculations that follow, it may be unclear why we insist on using the Squeeze Theorem before concluding that  is indeed 0. Review example 1.4 and section 1.1 to explain why we are being so fussy. |
| In exercises 5-32, evaluate the indicated limit.
|
| 5.  | 6.  |
| 7.  | 8.  |
| 9.  | 10.  |
| 11.  | 12.  |
| 13.  | 14.  |
| 15.  | 16.  |
| 17.  | 18.  |
| 19.  | 20.  |
| 21.  | 22.  |
| 23.  | 24. 
|
| 25.  , where 
|
| 26.  , where 
|
| 27.  , where 
|
| 28.  , where 
|
| 29.  , where 
|
| 30.  , where 
|
| 31.  , where 
|
| 32.  , where 
|
| 33. Use numerical and graphical evidence to conjecture the value of  . Use the Squeeze Theorem to prove that you are correct: identify the functions f (x) and h(x) , show graphically that f (x) x2sin (1/x) h(x) , and justify  . |
| 34. Why can't you use the Squeeze Theorem as in exercise 33 to prove that  ? Explore this limit graphically. |
| 35. Use the Squeeze Theorem to prove that  . Identify the functions f (x) and h(x) , show graphically that f (x)  cos 2(1/x) h(x) , and justify  and  . |
| 36. Suppose that f (x) is bounded: that is, there exist constants m and n such that m f (x) n for all x. Use the Squeeze Theorem to prove that  . |
| In exercises 37-40, either find the limit or explain why it does not exist.
|
| 37.  | 38.  |
| 39.  | 40. 
|
| 41. Given that  , quickly evaluate
 . |
| 42. Given that  , quickly evaluate  . |
| 43. Suppose  for polynomials g(x) and
h(x). Explain why  and determine  . |
| 44. Explain how to determine  if g and h are polynomials
and 
|
| 45. Evaluate each limit and justify each step by citing the appropriate theorem(s). |
|
(a)  | (b) |  |
|
| 46. Evaluate each limit and justify each step by citing the appropriate theorem(s). |
|
(a)  | (b) |  |
|
| In exercises 47-50, use the given position
function
f (t ) to find the velocity at time t =
a.
|
| 47. f (t ) = t 2 + 2, a = 2 | 48. f (t ) = t 2 + 2, a = 0 |
| 49. f (t ) = t 3, a = 0 | 50. f (t ) = t 3, a = 1
|
| 51. In
Chapter 2, the slope of the tangent line to the curve y =
at x = 1 is defined by  . Compute the
slope m. Graph y = and the line with slope m through the point (1, 1). |
| 52. In
Chapter 2, an alternative form for the limit in exercise 51 is given by  Compute this limit. |
| In exercises 53-56, use numerical evidence to conjecture the value of the limit if it exists. Check your answer with your CAS. If you disagree, which one of you is correct?
|
| 53.  | 54.  |
| 55.  | 56. 
|
| 57. Assume that  . Use
Theorem 2.3 to prove that  . Also, show that  . |
| 58. How did you work exercise 57? You probably used
Theorem 2.3 to work from  to  , and then used  to get  . Going one step at a time, we should be able to reach  for any positive integer n. This is the idea of mathematical induction. Formally, we need to show the result is true for a specific value of n = n0 (Corollary 2.1 does that for n0 = 2 ), then assume the result is true for a general n = k n0. If we show that we can get from the result being true for n = k to the result being true for n = k + 1 , we have proved that the result is true for any positive integer n. In one sentence, explain why this is true. Use this technique to prove that  for any positive integer n. |
| 59. Find all the errors in the following incorrect string of equalities:
|
| 60. Find all the errors in the following incorrect string of equalities:
|
| 61. Give an example of functions f (x) and g(x) such that  exists but  and  do not exist. |
| 62. Give an example of functions f (x) and g(x) such that  exists but at least one of  and  does not exist. |
| 63. If  exists and  does not exist, is it always true that  does not exist? Explain. |
| 64. Is the following true or false? If  does not exist, then  does not exist. Explain. |
| 65. Suppose a state's income tax code states the tax liability on x dollars of taxable income is given by
|
| Compute  ; why is this good? Compute  ; why is this bad? |
| 66. Suppose a state's income tax code states that tax liability is 12% on the first $20,000 of taxable earnings and 16% on the remainder. Find constants a and b for the tax
function 
such that  and  exists. Why is it important for these limits to exist? |
| 67. The greatest integer function is denoted by f (x) = [x] and equals the greatest integer which is less than or equal to x. Thus, [2.3] = 2, [ - 1.2] = - 2 and [3] = 3. In spite of this last fact, show that  does not exist. |
| 68. Investigate the existence of (a)  , (b)  , (c)  . |
 | 69. The value x = 0 is called a zero of multiplicity n (n 1 ) for the
function f (x) if  exists and is nonzero but  . Show that x = 0 is a zero of multiplicity 2 for x2, , x = 0 is a zero of multiplicity 3 for x3 and x = 0 is a zero of multiplicity 4 for x4. For polynomials, what does multiplicity describe? The reason the definition is not as straightforward as we might like is so that it can apply to nonpolynomial functions, as well. Find the multiplicity of x = 0 for f (x) = sin x ; f (x) = xsin x ; f (x) = sin x2. If you know that x = 0 is a zero of multiplicity m for f (x) and multiplicity n for g(x) , what can you say about the multiplicity of x = 0 for f (x) + g(x) ? f (x) g(x) ? f (g(x)) ? |
| | 70. We have conjectured that  . Using graphical and numerical evidence, conjecture the value of  ,  ,  and  . In general, conjecture the value of  for any
constant c. Given that  for any
constant c 0 , prove that your conjecture is correct. |
