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Limits and Continuity

Limits and Continuity. Chapter 2:. Aim #2.2 How do we solve limits as x approaches +/- infinity?. We use infinity when the values in its domain or range outgrow all finite bounds . Graph y = 1/x What happens as x approaches to + infinity? What happens as x approaches to - infinity?.

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Limits and Continuity

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  1. Limits and Continuity Chapter 2:

  2. Aim #2.2 How do we solve limits as x approaches +/- infinity? • We use infinity when the values in its domain or range outgrow all finite bounds. • Graph y = 1/x • What happens as x approaches to + infinity? • What happens as x approaches to - infinity?

  3. Definition: Horizontal Asymptote

  4. Example 1: Looking for Horizontal Asymptotes

  5. Graph the following f(x)

  6. Theorem 5

  7. Example 5: Using the Theorem 5

  8. Infinite Limits as x ---a • Graph the function f(x)= 1/x. What happens as x approaches 0 from the right? • From the left? What does y approach?

  9. Definition: Vertical Asymptotes

  10. Example 4: Finding the Vertical Asymptote What does f(x) approach as x approaches 0 from the right? From the left?

  11. Example 5: Finding Vertical Asymptotes • What are the vertical asymptotes of • f(x)= tan x?

  12. End Behavior Models • For numerically large values of x, we can sometimes model the behavior of a complicated function by a simpler one that acts virtually the same way.

  13. Example 6:

  14. Definition End Behavior Models

  15. Example 7:

  16. Seeing Limits as x approaches +/- infinity • We can investigate the graph of y =f(x) as x approaches +/- infinity by investigating the graph of y =f(1/x) as x approaches infinity.

  17. Example 9: Using Substitution

  18. Summary: Answer in complete sentences. • Explain why there is no value L for which lim as x approaches infinity for sin x = L

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