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Math 180

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  1. Math 180 Packet #2 Limits (Numeric and Algebraic)

  2. We can find limits given the graph of a function. But what if we don’t know the graph? One way to estimate the limit is numerically (that is, plug in numbers).

  3. We can find limits given the graph of a function. But what if we don’t know the graph? One way to estimate the limit is numerically (that is, plug in numbers).

  4. Ex 1.Find

  5. Ex 1.Find

  6. Ex 1.Find

  7. Ex 1.Find

  8. Ex 1.Find

  9. Ex 1.Find

  10. Ex 2.Find

  11. Ex 2.Find

  12. Ex 2.Find

  13. Ex 2.Find

  14. Ex 2.Find

  15. Ex 2.Find

  16. Ex 3.Find

  17. Ex 3.Find

  18. Ex 3.Find

  19. Ex 3.Find

  20. Ex 3.Find

  21. Ex 3.Find

  22. Making tables of numbers is cumbersome. Let’s transition to doing the analysis without a table.

  23. Ex 4.Find Ex 5.Find Ex 6.Find

  24. Ex 7.Find Ex 8.Find Ex 9.Find

  25. Now let’s get some building blocks for evaluating limits algebraically.

  26. Note: In general, ______ and ________.

  27. Note: In general, ______ and ________.

  28. Note: In general, ______ and ________. • ex: ____________

  29. Note: In general, ______ and ________. • ex: ____________

  30. Note: In general, ______ and ________. • ex: ____________

  31. The Limit Laws If and both exist, then the following laws are true:

  32. ex: Find using the Limit Laws.

  33. ex: Find using the Limit Laws.

  34. ex: Find using the Limit Laws.

  35. ex: Find using the Limit Laws.

  36. ex: Find using the Limit Laws.

  37. ex: Find using the Limit Laws.

  38. Note: If and are polynomials, then…

  39. Ex 10. Find

  40. Ex 10. Find

  41. What happens when both the numerator and denominator go to 0?

  42. Consider . What is ? ________________

  43. Consider . What is ? ________________ undefined

  44. Consider . What is ? ________________ undefined (Well, technically indeterminate…)

  45. Now what happens to as gets close to 1?

  46. Now what happens to as gets close to 1?

  47. Now what happens to as gets close to 1?

  48. Now what happens to as gets close to 1? It looks like approaches _____ as approaches 1.

  49. Now what happens to as gets close to 1? 2 It looks like approaches _____ as approaches 1.

  50. Now what happens to as gets close to 1? 2 It looks like approaches _____ as approaches 1. In other words: