Lesson 2: The Concept of Limit

1,411 views
1,313 views

Published on

Published in: Education
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
1,411
On SlideShare
0
From Embeds
0
Number of Embeds
32
Actions
Shares
0
Downloads
84
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

Lesson 2: The Concept of Limit

  1. 1. Section 1.3 The Concept of Limit V63.0121.002.2010Su, Calculus I New York University May 18, 2010 Announcements WebAssign Class Key: nyu 0127 7953 Office Hours: MR 5:00–5:45, TW 7:50–8:30, CIWW 102 (here) Quiz 1 Thursday on 1.1–1.4 . . . . . .
  2. 2. Announcements WebAssign Class Key: nyu 0127 7953 Office Hours: MR 5:00–5:45, TW 7:50–8:30, CIWW 102 (here) Quiz 1 Thursday on 1.1–1.4 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 2 / 32
  3. 3. Objectives Understand and state the informal definition of a limit. Observe limits on a graph. Guess limits by algebraic manipulation. Guess limits by numerical information. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 3 / 32
  4. 4. Last Time Key concept: function Properties of functions: domain and range Kinds of functions: linear, polynomial, power, rational, algebraic, transcendental. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 4 / 32
  5. 5. Limit . . . . . .
  6. 6. Zeno's Paradox That which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics VI:9, 239b10) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 5 / 32
  7. 7. Outline Heuristics Errors and tolerances Examples Pathologies Precise Definition of a Limit . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 6 / 32
  8. 8. Heuristic Definition of a Limit Definition We write lim f(x) = L x→a and say “the limit of f(x), as x approaches a, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 7 / 32
  9. 9. Outline Heuristics Errors and tolerances Examples Pathologies Precise Definition of a Limit . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 8 / 32
  10. 10. The error-tolerance game A game between two players to decide if a limit lim f(x) exists. x→a Step 1 Player 1 proposes L to be the limit. Step 2 Player 2 chooses an “error” level around L: the maximum amount f(x) can be away from L. Step 3 Player 1 looks for a “tolerance” level around a: the maximum amount x can be from a while ensuring f(x) is within the given error of L. The idea is that points x within the tolerance level of a are taken by f to y-values within the error level of L, with the possible exception of a itself. If Player 1 can do this, he wins the round. If he cannot, he loses the game: the limit cannot be L. Step 4 Player 2 go back to Step 2 with a smaller error. Or, he can give up and concede that the limit is L. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 9 / 32
  11. 11. The error-tolerance game L . . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  12. 12. The error-tolerance game L . . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  13. 13. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  14. 14. The error-tolerance game T . his tolerance is too big L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  15. 15. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  16. 16. The error-tolerance game S . till too big L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  17. 17. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  18. 18. The error-tolerance game T . his looks good L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  19. 19. The error-tolerance game S . o does this L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  20. 20. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. If Player 2 shrinks the error, Player 1 can still win. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  21. 21. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. If Player 2 shrinks the error, Player 1 can still win. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
  22. 22. Outline Heuristics Errors and tolerances Examples Pathologies Precise Definition of a Limit . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 11 / 32
  23. 23. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  24. 24. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  25. 25. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. Step 2 Player 2: I challenge you to make x2 within 0.01 of 0. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  26. 26. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. Step 2 Player 2: I challenge you to make x2 within 0.01 of 0. Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so a tolerance of 0.1 fits your error of 0.01. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  27. 27. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. Step 2 Player 2: I challenge you to make x2 within 0.01 of 0. Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so a tolerance of 0.1 fits your error of 0.01. Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  28. 28. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. Step 2 Player 2: I challenge you to make x2 within 0.01 of 0. Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so a tolerance of 0.1 fits your error of 0.01. Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0? Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so a tolerance of 0.01 fits your error of 0.0001. … . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  29. 29. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. Step 2 Player 2: I challenge you to make x2 within 0.01 of 0. Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so a tolerance of 0.1 fits your error of 0.01. Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0? Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so a tolerance of 0.01 fits your error of 0.0001. … Can you convince Player 2 that Player 1 can win every round? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  30. 30. Playing the Error-Tolerance game with x2 Example Find lim x2 if it exists. x→0 Solution Step 1 Player 1: I claim the limit is zero. Step 2 Player 2: I challenge you to make x2 within 0.01 of 0. Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so a tolerance of 0.1 fits your error of 0.01. Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0? Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so a tolerance of 0.01 fits your error of 0.0001. … Can you convince Player 2 that Player 1 can win every round? Yes, by setting the tolerance equal to the square root of the error, Player 1 can always win. Player 2 should give up and concede that the limit is 0. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
  31. 31. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  32. 32. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  33. 33. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  34. 34. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  35. 35. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  36. 36. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  37. 37. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  38. 38. Graphical version of the E-T game with x2 . . y . . . x . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  39. 39. Graphical version of the E-T game with x2 . . y . . . x . . No matter how small an error band Player 2 picks, Player 1 can find a fitting tolerance band. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
  40. 40. Limit of a piecewise function Example |x| Find lim if it exists. x→0 x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 14 / 32
  41. 41. Limit of a piecewise function Example |x| Find lim if it exists. x→0 x Solution The function can also be written as { |x| 1 if x > 0; = x −1 if x < 0 What would be the limit? . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 14 / 32
  42. 42. The E-T game with a piecewise function y . . . . . 1 . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  43. 43. The E-T game with a piecewise function y . . . . . 1 I . think the limit is 1 . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  44. 44. The E-T game with a piecewise function y . . . . . 1 I . think the limit is 1 . . .. x C . an you fit an error of 0.5? . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  45. 45. The E-T game with a piecewise function y . . . . . 1 H . ow about this for a tolerance? . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  46. 46. The E-T game with a piecewise function y . . . . . 1 H . ow about this for a tolerance? . . .. x . No. Part of graph inside . 1. − blue is not inside green . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  47. 47. The E-T game with a piecewise function y . . . . . 1 O . h, I guess the limit isn’t 1 . . .. x . No. Part of graph inside . 1. − blue is not inside green . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  48. 48. The E-T game with a piecewise function y . . . . . 1 . think the limit is −1 I . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  49. 49. The E-T game with a piecewise function y . . . . . 1 . think the limit is −1 I . . .. x C . an you fit an error of 0.5? . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  50. 50. The E-T game with a piecewise function y . . . . . 1 H . ow about this for a tolerance? . . .. x C . an you fit an error of 0.5? . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  51. 51. The E-T game with a piecewise function y . . . No. Part of . graph inside . . 1 blue is not inside . ow about this for a tolerance? green H . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  52. 52. The E-T game with a piecewise function y . . . No. Part of . graph inside . . 1 blue is not inside . h, I guess the limit isn’t −1 O green . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  53. 53. The E-T game with a piecewise function y . . . . . 1 I . think the limit is 0 . . .. x . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  54. 54. The E-T game with a piecewise function y . . . . . 1 I . think the limit is 0 . . .. x C . an you fit an error of 0.5? . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  55. 55. The E-T game with a piecewise function y . . . . . 1 H . ow about this for a tolerance? . . .. x C . an you fit an error of 0.5? . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  56. 56. The E-T game with a piecewise function y . . . . . 1 H . ow about this for a tolerance? . . .. x . No. None of . 1. − graph inside blue is inside green . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  57. 57. The E-T game with a piecewise function y . . . . . 1 . . Oh, I guess the . .. x limit isn’t 0 . No. None of . 1. − graph inside blue is inside green . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  58. 58. The E-T game with a piecewise function y . . . . . 1 . I give up! I . guess there’s . .. x no limit! . 1. − . . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
  59. 59. One-sided limits Definition We write lim f(x) = L x→a+ and say “the limit of f(x), as x approaches a from the right, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a and greater than a. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 16 / 32
  60. 60. One-sided limits Definition We write lim f(x) = L x→a− and say “the limit of f(x), as x approaches a from the left, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a and less than a. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 16 / 32
  61. 61. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  62. 62. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  63. 63. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  64. 64. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  65. 65. The error-tolerance game on the right y . . . 1 . x . . All of graph in- . 1. − side blue is in- side green . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  66. 66. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  67. 67. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  68. 68. The error-tolerance game on the right y . . . 1 . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  69. 69. The error-tolerance game on the right y . . All of graph in- . . 1 side blue is in- side green . x . . 1. − . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  70. 70. The error-tolerance game on the right y . . All of graph in- . . 1 side blue is in- side green . x . . 1. − So lim+ f(x) = 1 and lim f(x) = −1 x→0 x→0− . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
  71. 71. Limit of a piecewise function Example |x| Find lim if it exists. x→0 x Solution The function can also be written as { |x| 1 if x > 0; = x −1 if x < 0 What would be the limit? The error-tolerance game fails, but lim f(x) = 1 lim f(x) = −1 x→0+ x→0− . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 18 / 32
  72. 72. Another Example Example 1 Find lim+ if it exists. x→0 x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 19 / 32
  73. 73. The error-tolerance game with lim (1/x) x→0 y . .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  74. 74. The error-tolerance game with lim (1/x) x→0 y . .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  75. 75. The error-tolerance game with lim (1/x) x→0 y . .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  76. 76. The error-tolerance game with lim (1/x) x→0 y . . The graph escapes the green, so no good .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  77. 77. The error-tolerance game with lim (1/x) x→0 y . .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  78. 78. The error-tolerance game with lim (1/x) x→0 y . E . ven worse! .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  79. 79. The error-tolerance game with lim (1/x) x→0 y . . The limit does not ex- ist because the func- tion is unbounded near 0 .?. L . x . 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
  80. 80. Another (Bad) Example: Unboundedness Example 1 Find lim+ if it exists. x→0 x Solution The limit does not exist because the function is unbounded near 0. Later we will talk about the statement that 1 lim+ = +∞ x→0 x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 21 / 32
  81. 81. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 22 / 32
  82. 82. Function values x π/x sin(π/x) . /2 π . 1 π 0 1/2 2π 0 1/k kπ 0 2 π/2 1 2/5 5π/2 1 . . π . .. 0 2/9 9π/2 1 2/13 13π/2 1 2/3 3π/2 −1 2/7 7π/2 −1 . 2/11 11π/2 −1 3 . π/2 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 23 / 32
  83. 83. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
  84. 84. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x f(x) = 0 when x = f(x) = 1 when x = f(x) = −1 when x = . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
  85. 85. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x 1 f(x) = 0 when x = for any integer k k f(x) = 1 when x = f(x) = −1 when x = . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
  86. 86. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x 1 f(x) = 0 when x = for any integer k k 2 f(x) = 1 when x = for any integer k 4k + 1 f(x) = −1 when x = . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
  87. 87. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x 1 f(x) = 0 when x = for any integer k k 2 f(x) = 1 when x = for any integer k 4k + 1 2 f(x) = −1 when x = for any integer k 4k − 1 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
  88. 88. Weird, wild stuff continued Here is a graph of the function: y . . . 1 . x . . 1. − There are infinitely many points arbitrarily close to zero where f(x) is 0, or 1, or −1. So the limit cannot exist. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 25 / 32
  89. 89. Outline Heuristics Errors and tolerances Examples Pathologies Precise Definition of a Limit . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 26 / 32
  90. 90. What could go wrong? Summary of Limit Pathologies How could a function fail to have a limit? Some possibilities: left- and right- hand limits exist but are not equal The function is unbounded near a Oscillation with increasingly high frequency near a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 27 / 32
  91. 91. Meet the Mathematician: Augustin Louis Cauchy French, 1789–1857 Royalist and Catholic made contributions in geometry, calculus, complex analysis, number theory created the definition of limit we use today but didn’t understand it . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 28 / 32
  92. 92. Outline Heuristics Errors and tolerances Examples Pathologies Precise Definition of a Limit . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 29 / 32
  93. 93. Precise Definition of a Limit No, this is not going to be on the test Let f be a function defined on an some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write lim f(x) = L, x→a if for every ε > 0 there is a corresponding δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 30 / 32
  94. 94. The error-tolerance game = ε, δ L . . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  95. 95. The error-tolerance game = ε, δ L . +ε L . . −ε L . a . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  96. 96. The error-tolerance game = ε, δ L . +ε L . . −ε L . . − δ. . + δ a aa . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  97. 97. The error-tolerance game = ε, δ T . his δ is too big L . +ε L . . −ε L . . − δ. . + δ a aa . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  98. 98. The error-tolerance game = ε, δ L . +ε L . . −ε L . . −. δ δ a . a+ a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  99. 99. The error-tolerance game = ε, δ T . his δ looks good L . +ε L . . −ε L . . −. δ δ a . a+ a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  100. 100. The error-tolerance game = ε, δ S . o does this δ L . +ε L . . −ε L . . .− δ δ aa .+ a . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
  101. 101. Summary y . Fundamental Concept: . . 1 limit Error-Tolerance game . x . gives a methods of arguing limits do or do not exist Limit FAIL: jumps, . 1. − unboundedness, sin(π/x) FAIL . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 32 / 32

×