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# Lesson 3: The Limit of a Function

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Limits are where algebra ends and calculus begins.

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### Lesson 3: The Limit of a Function

1. 1. Section 1.3 The Limit of a Function V63.0121, Calculus I January 26–27, 2009 Announcements Oﬃce Hours: MW 1:30–3:00, TR 1:00–2:00 (WWH 718) Blackboard operational HW due Wednesday, ALEKS initial due Friday
2. 2. Limit
3. 3. 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)
4. 4. Outline The Concept of Limit Heuristics Errors and tolerances Examples Pathologies
5. 5. 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.
6. 6. The error-tolerance game A game between two players to decide if a limit lim f (x) exists. x→a Player 1: Choose L to be the limit. Player 2: Propose an “error” level around L. Player 1: Choose a “tolerance” level around a so that x-points within that tolerance level are taken to y -values within the error level. If Player 1 can always win, lim f (x) = L. x→a
7. 7. The error-tolerance game L a
8. 8. The error-tolerance game L a
9. 9. 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.
10. 10. The error-tolerance game This 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.
11. 11. 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.
12. 12. The error-tolerance game Still 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.
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.
14. 14. The error-tolerance game This 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.
15. 15. The error-tolerance game So 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.
16. 16. 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.
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. If Player 2 shrinks the error, Player 1 can still win.
18. 18. Example Find lim x 2 if it exists. x→0
19. 19. Example Find lim x 2 if it exists. x→0 Solution By setting tolerance equal to the square root of the error, we can guarantee to be within any error.
20. 20. Example |x| Find lim if it exists. x→0 x
21. 21. 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?
22. 22. The error-tolerance game y 1 x −1
23. 23. The error-tolerance game y 1 x −1
24. 24. The error-tolerance game y 1 x −1
25. 25. The error-tolerance game y 1 x −1
26. 26. The error-tolerance game y 1 x Part of graph in- −1 side blue is not inside green
27. 27. The error-tolerance game y 1 x −1
28. 28. The error-tolerance game y 1 x −1
29. 29. The error-tolerance game y Part of graph in- side blue is not 1 inside green x −1
30. 30. The error-tolerance game y Part of graph in- side blue is not 1 inside green x −1 These are the only good choices; the limit does not exist.
31. 31. 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 (on either side of a) and greater than a.
32. 32. 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 (on either side of a) and less than a.
33. 33. 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+
34. 34. Example 1 Find lim+ if it exists. x x→0
35. 35. The error-tolerance game y L? x 0
36. 36. The error-tolerance game y L? x 0
37. 37. The error-tolerance game y L? x 0
38. 38. The error-tolerance game y The graph escapes the green, so no good L? x 0
39. 39. The error-tolerance game y L? x 0
40. 40. The error-tolerance game y Even worse! L? x 0
41. 41. The error-tolerance game y The limit does not exist because the function is unbounded near 0 L? x 0
42. 42. Example 1 Find lim+ if it exists. x x→0 Solution The limit does not exist because the function is unbounded near 0. Next week we will understand the statement that 1 lim+ = +∞ x x→0
43. 43. Example π Find lim sin if it exists. x x→0
44. 44. y 1 x −1
45. 45. What could go wrong? 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
46. 46. 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
47. 47. Precise Deﬁnition of a Limit 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.
48. 48. The error-tolerance game = ε, δ L a
49. 49. The error-tolerance game = ε, δ L+ε L L−ε a
50. 50. The error-tolerance game = ε, δ L+ε L L−ε a − δaa + δ
51. 51. The error-tolerance game = ε, δ This δ is too big L+ε L L−ε a − δaa + δ
52. 52. The error-tolerance game = ε, δ L+ε L L−ε a −aδ δ a+
53. 53. The error-tolerance game = ε, δ This δ looks good L+ε L L−ε a −aδ δ a+
54. 54. The error-tolerance game = ε, δ So does this δ L+ε L L−ε aa a δ δ − +