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Limits are where algebra ends and calculus begins.

Limits are where algebra ends and calculus begins.

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- 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. Limit
- 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. Outline The Concept of Limit Heuristics Errors and tolerances Examples Pathologies
- 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. 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. The error-tolerance game L a
- 8. The error-tolerance game L a
- 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. 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. 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. 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. 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. 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. 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. 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. 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. Example Find lim x 2 if it exists. x→0
- 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. Example |x| Find lim if it exists. x→0 x
- 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. The error-tolerance game y 1 x −1
- 23. The error-tolerance game y 1 x −1
- 24. The error-tolerance game y 1 x −1
- 25. The error-tolerance game y 1 x −1
- 26. The error-tolerance game y 1 x Part of graph in- −1 side blue is not inside green
- 27. The error-tolerance game y 1 x −1
- 28. The error-tolerance game y 1 x −1
- 29. The error-tolerance game y Part of graph in- side blue is not 1 inside green x −1
- 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. 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. 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. 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. Example 1 Find lim+ if it exists. x x→0
- 35. The error-tolerance game y L? x 0
- 36. The error-tolerance game y L? x 0
- 37. The error-tolerance game y L? x 0
- 38. The error-tolerance game y The graph escapes the green, so no good L? x 0
- 39. The error-tolerance game y L? x 0
- 40. The error-tolerance game y Even worse! L? x 0
- 41. The error-tolerance game y The limit does not exist because the function is unbounded near 0 L? x 0
- 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. Example π Find lim sin if it exists. x x→0
- 44. y 1 x −1
- 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. 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. 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. The error-tolerance game = ε, δ L a
- 49. The error-tolerance game = ε, δ L+ε L L−ε a
- 50. The error-tolerance game = ε, δ L+ε L L−ε a − δaa + δ
- 51. The error-tolerance game = ε, δ This δ is too big L+ε L L−ε a − δaa + δ
- 52. The error-tolerance game = ε, δ L+ε L L−ε a −aδ δ a+
- 53. The error-tolerance game = ε, δ This δ looks good L+ε L L−ε a −aδ δ a+
- 54. The error-tolerance game = ε, δ So does this δ L+ε L L−ε aa a δ δ − +

Full NameComment goes here.Matthew Leingang, Clinical Associate Professor of Mathematics at New York University Latest version: http://www.slideshare.net/leingang/lesson-3-the-limit-of-a-function-slidesbeautygurlz3 years agoPenny Ng, Student at Universiti Tunku Abdul Rahman (UTAR) 3 years agothessanie3 years agoDarren Kuropatwa, Educator at ∞ß Tagscalculuslimits5 years ago