Chapter 3 Section 5

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Chapter 3Section 5

• Limits at Infinity
• (Horizontal Asymptotes)

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Within this lesson

• We will learn how to
• Find the horizontal asymptotes of a function.
• Appropriately support our work using Calculus
  notations.

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How do we use Calculus notation to justify a
horizontal asymptote?

• Find the horizontal asymptote(s) of the
  following

As x approaches infinity, y approaches 1.
As x approaches negative infinity, y approaches
1.
So Horizontal Asymptote at y 1.
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Example 1

• Graph on your calculator and
  observe the behavior as .
• Fill in the following table with your results
• Whats your conclusion and how do you support it
  with proper notation?

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Example 2

• Evaluate with your graphing calculator
• How can you find your answer analytically?

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Example 3

• Use four different analytic methods to find
  .
• 1. Graphically

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Example 3

• 2. Divide all terms by highest power

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Example 3

• 3. Compare exponents between numerator and
  denominator

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LHopitals Rule
Named after the 17th-century French mathematician
Guillaume de l'Hôpital

• Applies to quotients whose limits are
  indeterminate forms.
• Used to convert an indeterminate form to a
  determinate form, which allows for easier
  computation of the limit.

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LHopitals Rule

• In symbols
• This is what we wish the quotient rule really was
  the derivative of the numerator over the
  derivative of the denominator.
• Continue performing LHopitals Rule until you
  can plug in and wont get an indeterminate form.

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Example 3

• 4. LHopitals Rule

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Example 4

• Find all horizontal asymptotes of the following

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Example 5

• Find all horizontal asymptotes for the following

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A Summary

• If degree of the numerator is smaller than the
  degree of the denominator
• BOTTOM HEAVY when infinity is plugged in
• If degree of the numerator is the same as the
  degree of the denominator
• Take the ratio of the leading coefficients.
• If the degree of the numerator is larger than the
  degree of the demoninator
• TOP HEAVY when infinity is plugged in

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Homework

• 3.5
• p. 193
• 1 4, 6, 9, 12, 14,
• 16, 18, 34, 42, 65