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The Idea of ‘Limits’

The Idea of ‘Limits’

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The Idea of ‘Limits’

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  1. 1.4 The Idea of ‘Limits’

  2. Average velocities over time:

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

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

  5. 1.5 The Limit of a Function

  6. Definition and Notation

  7. Finding Limits using Graphs:

  8. Finding Limits using Tables:

  9. One-Sided Limits:

  10. Example Table

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

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

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

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

  15. Example:

  16. Example:

  17. All 4 possibilities:

  18. Vertical Asymptote:

  19. Example:

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

  21. Example:

  22. Horizontal Asymptotes:

  23. Example:

  24. Example:

  25. Examples:

  26. Examples: