chapter_02

# chapter_02 - 11:43 L24-ch02 Sheet number 1 Page number 49...

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January 27, 2005 11:43 L24-ch02 Sheet number 1 Page number 49 black 49 CHAPTER 2 Limits and Continuity EXERCISE SET 2.1 1. (a) 0 (b) 0 (c) 0 (d) 3 2. (a) + (b) + (c) + (d) undef 3. (a) −∞ (b) −∞ (c) −∞ (d) 1 4. (a) 1 (b) −∞ (c) does not exist (d) 2 5. for all x 0 6 = 4 6. for all x 0 6 = 6 , 3 13. (a) 2 1 . 5 1 . 1 1 . 01 1 . 001 0 0 . 5 0 . 9 0 . 99 0 . 999 0 . 1429 0 . 2105 0 . 3021 0 . 3300 0 . 3330 1 . 0000 0 . 5714 0 . 3690 0 . 3367 0 . 3337 1 0 02 The limit is 1 / 3. (b) 2 1 . 5 1 . 1 1 . 01 1 . 001 1 . 0001 0 . 4286 1 . 0526 6 . 344 66 . 33 666 . 3 6666 . 3 50 0 12 The limit is + . (c) 0 0 . 5 0 . 9 0 . 99 0 . 999 0 . 9999 1 1 . 7143 7 . 0111 67 . 001 667 . 0 6667 . 0 0 –50 01 The limit is −∞ .

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January 27, 2005 11:43 L24-ch02 Sheet number 2 Page number 50 black 50 Chapter 2 14. (a) 0 . 25 0 . 1 0 . 001 0 . 0001 0 . 0001 0 . 001 0 . 1 0 . 25 0 . 5359 0 . 5132 0 . 5001 0 . 5000 0 . 5000 0 . 4999 0 . 4881 0 . 4721 0.6 0 –0.25 0.25 The limit is 1/2. (b) 0 . 25 0 . 1 0 . 001 0 . 0001 8 . 4721 20 . 488 2000 . 5 20001 100 0 0 0.25 The limit is + . (c) 0 . 25 0 . 1 0 . 001 0 . 0001 7 . 4641 19 . 487 1999 . 5 20000 0 –100 –0.25 0 The limit is −∞ . 15. (a) 0 . 25 0 . 1 0 . 001 0 . 0001 0 . 0001 0 . 001 0 . 1 0 . 25 2 . 7266 2 . 9552 3 . 0000 3 . 0000 3 . 0000 3 . 0000 2 . 9552 2 . 7266 3 2 –0.25 0.25 The limit is 3.
January 27, 2005 11:43 L24-ch02 Sheet number 3 Page number 51 black Exercise Set 2.1 51 (b) 0 0 . 5 0 . 9 0 . 99 0 . 999 1 . 5 1 . 1 1 . 01 1 . 001 1 1 . 7552 6 . 2161 54 . 87 541 . 1 0 . 1415 4 . 536 53 . 19 539 . 5 60 –60 –1.5 0 The limit does not exist. 16. (a) 0 0 . 5 0 . 9 0 . 99 0 . 999 1 . 5 1 . 1 1 . 01 1 . 001 1 . 5574 1 . 0926 1 . 0033 1 . 0000 1 . 0000 1 . 0926 1 . 0033 1 . 0000 1 . 0000 1.5 1 –1.5 0 The limit is 1. (b) 0 . 25 0 . 1 0 . 001 0 . 0001 0 . 0001 0 . 001 0 . 1 0 . 25 1 . 9794 2 . 4132 2 . 5000 2 . 5000 2 . 5000 2 . 5000 2 . 4132 1 . 9794 2.5 2 –0.25 0.25 The limit is 5/2. 17. m sec = x 2 1 x +1 = x 1 which gets close to 2as x gets close to 1, thus y 1= 2( x +1)or y = 2 x 1 18. m sec = x 2 x = x which gets close to 0 as x gets close to 0 (doh!), thus y =0 19. m sec = x 4 1 x 1 = x 3 + x 2 + x + 1 which gets close to 4 as x gets close to 1, thus y 1=4( x 1) or y =4 x 3 20. m sec = x 4 1 x = x 3 x 2 + x 1 which gets close to 4as x gets close to 1, thus y 4( x +1) or y = 4 x 3

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January 27, 2005 11:43 L24-ch02 Sheet number 4 Page number 52 black 52 Chapter 2 21. (a) The limit appears to be 3. 3.5 2.5 11 (b) The limit appears to be 3. 3.5 2.5 –0.001 0.001 (c) The limit does not exist. 3.5 2.5 –0.000001 0.000001 22. (a) 0 . 01 0 . 001 0 . 0001 0 . 00001 1 . 666 0 . 1667 0 . 1667 0 . 17 (b) 0 . 000001 0 . 0000001 0 . 00000001 0 . 000000001 0 . 0000000001 0 0 0 0 0 (c) it can be misleading 23. (a) The plot over the interval [ a, a ] becomes subject to catastrophic subtraction if a is small enough (the size depending on the machine). (c) It does not. 24. (a) the mass of the object while at rest (b) the limiting mass as the velocity approaches the speed of light; the mass is unbounded 25. (a) The length of the rod while at rest (b) The limit is zero. The length of the rod approaches zero EXERCISE SET 2.2 1. (a) 6 (b) 13 (c) 8 (d) 16 (e) 2 (f) 1 / 2 (g) The limit doesn’t exist because the denominator tends to zero but the numerator doesn’t. (h) The limit doesn’t exist because the denominator tends to zero but the numerator doesn’t. 2. (a) 0 (b) The limit doesn’t exist because lim f doesn’t exist and lim g does. (c) 0 (d) 3 (e) 0 The limit doesn’t exist because the denominator tends to zero but the numerator doesn’t.
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chapter_02 - 11:43 L24-ch02 Sheet number 1 Page number 49...

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