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Section 1.2

Section 1.2. Introduction to Limits. Definition of Limit. If f ( x ) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f ( x ) as x approaches c is L . This is written as.

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Section 1.2

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  1. Section 1.2 Introduction to Limits

  2. Definition of Limit

  3. If f(x) becomes arbitrarily close to a • unique number L as x approaches c • from either side, the limit of f(x) as x • approaches c is L. This is written as

  4. One way to estimate the limit is numerically. That is , use a table to estimate the limit by looking at values on either side of the value of x in the limit.

  5. Example 1

  6. Use a table and the definition of a limit to estimate numerically the limit: 11.500 11.950 11.995 12.005 12.05 12.5

  7. Example 2

  8. Use a table and the definition of a limit to estimate numerically the limit: 5.998 5.9998 5.99998 6.00002 6.0002 6.002

  9. Example 3

  10. 1. Graph this function by hand.

  11. SOME EXAMPLES OF LIMITS THAT DO NOT EXIST

  12. Example 4

  13. Show that the limit does not exist. • 1. Graph the function by hand. • HINT: Make a table.

  14. Since f(x) as x approaches 1 from • the left is -1 and f(x) as x • approaches 1 from the right is 1, no • limit exists. • This is written as

  15. Example 5

  16. Discuss the existence of the limit: • 1. Graph the function by hand. • HINT: Make a table.

  17. As x approaches 0 from the left what happens to f(x)? f(x) increases without bound as x approaches 0 from the left. As x approaches 0 from the right what happens to f(x)? f(x) increases without bound as x approaches 0 from the right.

  18. Since f(x) increases without bound as x approaches 0, you can conclude that the limit does not exist. • This is written as

  19. Example 6

  20. Discuss the existence of the limit 1. Graph the function on a graphing calculator. Use -1.5 ≤ x ≤ 1.5 and -1.5 ≤ y ≤ 1.5 for your window.

  21. Since f(x) oscillates between -1 and 1 as • x approaches 0, the limit does not exist. • This is written as • Read Common Types of Behavior Associated with Nonexistence of a Limit on p. 51. • HW: pp. 54-56 (2-28 even) • Read pp. 59-61.

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