2. Packet #26 The Precise Definition of a Limit
Math 180
Packet #26
The Precise Definition of a Limit

3. ex: What is the distance between 1 and 3 on the number line?

4. ex: What is the distance between 1 and 3 on the number line?

5. ex: What is the distance between 1 and 3 on the number line?

6. ex: What is the distance between -2 and 3 on the number line?

7. ex: What is the distance between -2 and 3 on the number line?

8. ex: What is the distance between -2 and 3 on the number line?

9. ex: What is the distance between 𝑥 and 2 on the number line
ex: What is the distance between 𝑥 and 2 on the number line? Remember: distances are positive.

10. ex: What is the distance between 𝑥 and 2 on the number line
ex: What is the distance between 𝑥 and 2 on the number line? Remember: distances are positive.

11. ex: What is the distance between 𝑥 and 2 on the number line
ex: What is the distance between 𝑥 and 2 on the number line? Remember: distances are positive.

12. ex: What is the distance between 𝑥 and 2 on the number line
ex: What is the distance between 𝑥 and 2 on the number line? Remember: distances are positive.

13. ex: What is the distance between 𝑥 and 2 on the number line
ex: What is the distance between 𝑥 and 2 on the number line? Remember: distances are positive.

14. ex: What is the distance between 𝑥 and 2 on the number line
ex: What is the distance between 𝑥 and 2 on the number line? Remember: distances are positive.

15. ex: What do these expressions mean on the number line? 𝑥−3 𝑥+5 𝑥

16. ex: What do these expressions mean on the number line? 𝑥−3 𝑥+5 𝑥

17. ex: What do these expressions mean on the number line? 𝑥−3 𝑥+5 𝑥

18. ex: What do these expressions mean on the number line? 𝑥−3 𝑥+5 𝑥

19. ex: What do these expressions mean on the number line? 𝑥−3 𝑥+5 𝑥

20. ex: What do these expressions mean on the number line? 𝑥−3 𝑥+5 𝑥

21. ex: Solve 𝑥−2 =7.

22. ex: Solve 𝑥−2 =7.

23. ex: Solve 𝑥−2 =7.

24. ex: Solve 𝑥−2 =7.

25. ex: Solve 𝑥−2 =7.

26. ex: Solve 𝑥−2 =7.

27. ex: Solve 𝑥−2 <7. Show the solutions on the number line.

28. ex: Solve 𝑥−2 <7. Show the solutions on the number line.

29. ex: Solve 𝑥−2 <7. Show the solutions on the number line.

30. ex: Show 𝑥− 𝑥 0 <3 on the number line ( 𝑥 0 is a constant).

31. ex: Show 𝑥− 𝑥 0 <3 on the number line ( 𝑥 0 is a constant).

32. ex: Show 𝑥− 𝑥 0 <3 on the number line ( 𝑥 0 is a constant).

33. ex: Show 𝑥− 𝑥 0 <3 on the number line ( 𝑥 0 is a constant).

34. ex: Show 𝑥− 𝑥 0 <3 on the number line ( 𝑥 0 is a constant).

35. ex: Show 𝑥− 𝑥 0 <𝛿 on the number line.

36. ex: Show 𝑥− 𝑥 0 <𝛿 on the number line.

37. ex: Show 𝑥− 𝑥 0 <𝛿 on the number line.

38. ex: Show 𝑥− 𝑥 0 <𝛿 on the number line.

39. ex: Show 𝑥− 𝑥 0 <𝛿 on the number line.

42. Definition of Limit
Let 𝑓(𝑥) be defined on an open interval about 𝑥 0 , except possibly at 𝑥 0 itself. We say that the limit of 𝑓(𝑥) as 𝑥 approaches 𝑥 0 is the number 𝐿, and we write lim 𝑥→ 𝑥 0 𝑓(𝑥) =𝐿 if for every number 𝜖>0, there exists a corresponding number 𝛿>0 such that for all 𝑥, 0< 𝑥− 𝑥 0 <𝛿 ⇒ 𝑓 𝑥 −𝐿 <𝜖 . Note: ⇒ means “implies” (think: “if…then”)

43. Definition of Limit
Let 𝑓(𝑥) be defined on an open interval about 𝑥 0 , except possibly at 𝑥 0 itself. We say that the limit of 𝑓(𝑥) as 𝑥 approaches 𝑥 0 is the number 𝐿, and we write lim 𝑥→ 𝑥 0 𝑓(𝑥) =𝐿 if for every number 𝜖>0, there exists a corresponding number 𝛿>0 such that for all 𝑥, 0< 𝑥− 𝑥 0 <𝛿 ⇒ 𝑓 𝑥 −𝐿 <𝜖 . Note: ⇒ means “implies” (think: “if…then”)

44. Ex 1. Use the given graph of 𝑓 to find a number 𝛿 such that if 𝑥−2 <𝛿 then 𝑓 𝑥 −0.5 <0.25.

45. Ex 1. Use the given graph of 𝑓 to find a number 𝛿 such that if 𝑥−2 <𝛿 then 𝑓 𝑥 −0.5 <0.25.

46. Ex 1. Use the given graph of 𝑓 to find a number 𝛿 such that if 𝑥−2 <𝛿 then 𝑓 𝑥 −0.5 <0.25.

47. Ex 1. Use the given graph of 𝑓 to find a number 𝛿 such that if 𝑥−2 <𝛿 then 𝑓 𝑥 −0.5 <0.25.

48. Ex 1. Use the given graph of 𝑓 to find a number 𝛿 such that if 𝑥−2 <𝛿 then 𝑓 𝑥 −0.5 <0.25.

49. Ex 1. Use the given graph of 𝑓 to find a number 𝛿 such that if 𝑥−2 <𝛿 then 𝑓 𝑥 −0.5 <0.25.

50. Ex 1. Use the given graph of 𝑓 to find a number 𝛿 such that if 𝑥−2 <𝛿 then 𝑓 𝑥 −0.5 <0.25.

51. Ex 1. Use the given graph of 𝑓 to find a number 𝛿 such that if 𝑥−2 <𝛿 then 𝑓 𝑥 −0.5 <0.25.

52. Ex 2. For lim 𝑥→5 𝑥−1 =2, find a 𝛿>0 that works for 𝜖=1.

53. Ex 2. For lim 𝑥→5 𝑥−1 =2, find a 𝛿>0 that works for 𝜖=1.

54. Ex 2. For lim 𝑥→5 𝑥−1 =2, find a 𝛿>0 that works for 𝜖=1.

55. Ex 2. For lim 𝑥→5 𝑥−1 =2, find a 𝛿>0 that works for 𝜖=1.

56. Ex 2. For lim 𝑥→5 𝑥−1 =2, find a 𝛿>0 that works for 𝜖=1.

57. Ex 3. For lim 𝑥→−1 1/𝑥 =−1, find a 𝛿>0 that works for 𝜖=0.1.

58. Ex 4. Prove that lim 𝑥→1 5𝑥−3 =2 by using the 𝜖, 𝛿 definition of a limit.

59. Ex 5. Prove that lim 𝑥→2 𝑓(𝑥) =4 by using the 𝜖, 𝛿 definition of a limit if 𝑓 𝑥 = 𝑥 2 , 𝑥≠2 1, 𝑥=2

60. Note: Here is some mathematical shorthand that you might see: ∀ means “for every” or “for all” ∃ means “there exists” s.t. means “such that”.

61. So, the definition of the limit can be written compactly: lim 𝑥→ 𝑥 0 𝑓(𝑥) =𝐿 if ∀𝜖>0, ∃𝛿>0 s.t.∀𝑥, 0< 𝑥− 𝑥 0 <𝛿⇒ 𝑓 𝑥 −𝐿 <𝜖