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 | 1. In his 1726 masterpiece Mathematical Principles of Natural Philosophy, which introduces many of the fundamentals of calculus, Isaac Newton described the important limit  (which we study at length in |
| Chapter 2) as “the limit to which the ratios of quantities decreasing without limit do always converge, and to which they approach nearer than by any given difference, but never gobeyond, nor ever reach until the quantities vanish. ” If you ever get weary ofall the notation that we use in calculus, think of what itwould look like in words! Critique Newton's definition of limit, addressing the following questions in the process. What restrictions do the phrases “never go beyond” and “neverreach” put on the limit process? Give an example of a simple limit, not necessarily of the form  , that violates these restrictions. Give your own (English language) description of the limit, avoiding
restrictions such as Newton's. Why do mathematicians consider the  definition simple and elegant? |
| | 2. You have computed numerous limits before seeing the definition of limit. Explain how this definition changes and/or improves your understanding of the limit process. |
| | 3. Each word in the  definition is carefully chosen and precisely placed. Describe what is wrong with each of the following slightly incorrect “definitions” (use examples!):
 | (a) There exists  such that there exists a  > 0 such that if 0 < |x - a| < , then  .
| (b) For all  and for all > 0 , if 0 < |x - a| < , then  .
| (c) For all > 0 there exists  such that 0 < |x - a| < and  .
| | |
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| | 4. In order for the limit to exist, given every  we must be able to find a > 0 such that the if/then inequalities are true. To prove that the limit does not exist, we must find a particular  such that the if/then inequalities are not true for any choice of > 0. To understand the logic behind the swapping of the “for every” and “there exists” roles, draw an analogy with the following situation. Suppose the statement, “Everybody loves somebody” is true. If you wanted to verify the statement, why would you have to talk to every person on earth? But, suppose that the statement is not true. What would you have to do to disprove it? |
| In exercises 5-12, numerically and graphically determine a corresponding to the given  . Graph the function in the - window [x - range is - + ) and y - range is L - L + )] to verify that your choice works.
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| 5.  | 6.  |
| 7.  | 8.  |
| 9.  | 10.  |
| 11.  | 12. 
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| In exercises 13-24, symbolically find in terms of .
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| 13.  | 14.  |
| 15.  | 16.  |
| 17.  | 18.  |
| 19.  | 20. 
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| 21.  | 22.  |
| 23.  | 24. 
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| 25. Determine a formula for in terms of  for  (Hint: Use exercises 13-20). Does the formula depend on the value of a? Try to explain this answer graphically. |
| 26. Based on exercises 21 and 23, does the value of depend on the value of a for  ? Try to explain this graphically. |
| 27. Modify the  definition to define the one-sided limits
 and 
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| 28. Symbolically find the largest corresponding to  in the definition of  . Symbolically find the largest
corresponding to  in the definition of  .
Which could be used in the definition of  ? Briefly
explain. Then prove that  . |
| In exercises 29-34, find a corresponding to M = 100 or N - 100 (as appropriate) for each limit.
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| 29.  | 30.  |
| 31.  | 32.  |
| 33.  | 34. 
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| In exercises 35-40, find an M corresponding to  for each limit at infinity.
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| 35.  | 36.  |
| 37.  | 38.  |
| 39.  | 40. 
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| In exercises 41-48, prove that the limit is correct using the appropriate definition (assume that k is an integer.)
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| 41.  | 42.  |
| 43.  for k > 0 | 44.  for k > 0 |
| 45.  | 46. 
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| 47.  | 48.  |
| 49.  | 50. 
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| In exercises 51-54, identify a specific $epsilon;>0 for which no >0 exists to satisfy the definition of limit.
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| 51.  
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| 52.  
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| 53. 
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| 54. 
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| 55. A metal washer of (outer) radius r inches weighs 2r2 ounces. A company manufactures 2 - inch washers for different customers who have different error tolerances. If the customer demands a washer of weight  ounces, what is the error tolerance for the radius? That is, find such that a radius of r within the interval (2 - , 2 + ) guarantees a weight within  . |
| 56. A fiberglass company ships its glass as spherical marbles. If the volume of each marble has to be within  of  /6 , how close does the radius have to be to 1/2 ? |
 | 57. We hope that working through this section has provided you with extra insight into the limit process. However, we have not yet solved any problems we could not already solve in previous sections. We do so now, while investigating an unusual function. Recall that rational numbers can be written as fractions p/q, where p and q are integers. We will assume that p/q has been simplified by dividing out common factors (e.g., 1/2 and not 2/4 ). Define
 We will try to show that
 exists. Without graphics, we need a good definition
to answer this question. We know that f (2/3) = 1/3 , but recall that the limit is independent of the actual function value. We need to think about x 's close to 2/3. If such an x is irrational, f (x) = 0. A simple hypothesis would then be  .
We'll try this out for  . We would like to guarantee that  whenever  . Well, how many x's have a function value greater than 1/6? The only possible function values are 1/5, 1/4, 1/3, 1/2 and 1. The x's with function value 1/5 are 1/5, 2/5, 3/5, 4/5 and so on. The closest of these x's to 2/3 is 3/5. Find the closest x (not counting x = 2/3 ) to 2/3 with function value 1/4. Repeat for f (x) = 1/3 , f (x) = 1/2 and f (x) = 1. Out of all these closest x's, how close is the absolute closest? Choose to be this number, and argue that if  we are guaranteed that  . Argue that a similar process can find a for any  . |
