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Lesson 2: The Concept of Limit

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  • 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. 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. 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. 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. Limit . . . . . .
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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