The Idea of ‘Limits’

1. 1.4 The Idea of ‘Limits’

2. Average velocities over time:

4. “Instantaneous” velocity at one moment in time:

5. Limit of the Average velocity = the Instantaneous velocity: The Secant approaches the Tangent (average velocity) (instantaneous velocity)

6. 1.5 The Limit of a Function

7. Definition and Notation

8. Finding Limits using Graphs:

12. Finding Limits using Tables:

13. One-Sided Limits:

14. Example Table

15. The limit exists if and only if “left” limit = “right” limit

16. Example where limit does not exist: “Jump” behavior As x → 2: Left limit? Right limit?

17. Example where limit does not exist: undefined behavior near 0

18. Infinite Limits (When x approaches a number, f(x) approaches infinity)

19. Example:

21. Example:

23. All 4 possibilities:

24. Vertical Asymptote:

25. Example:

26. or Limits at Infinity (x approaches infinity, f(x) approaches a finite limit L, or infinity)

27. Example:

28. Horizontal Asymptotes:

30. Example:

31. Example:

32. Examples:

35. Examples: