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Indeterminate Forms and L’Hopital’s Rule
Section 8.7 Calculus AP/Dual, Revised ©2017 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Background L’Hôpital’s Rule was developed by his teacher Johann Bernoulli. Guillaume De L’Hôpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons. 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Review Evaluate 𝐥𝐢𝐦 𝒙→𝟐 𝒙 𝟐 −𝟒 𝒙−𝟐 using the limit process 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Video 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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Strategies of Solving Algebraically
Try direct substitution If direct substitution fails, try simplifying or factoring. If direct substitution fails, and there is a √ in the numerator, rationalize the numerator (multiply by the conjugate). 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Steps PLUG the equation first to determine whether it is indeterminate If 𝐥𝐢𝐦 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝟎 𝟎 𝒐𝒓 ∞ ∞ , then 𝐥𝐢𝐦 𝒙→𝒄 𝒇 ′ 𝒙 𝒈 ′ 𝒙 It is not Quotient Rule due to the limit If it’s no longer indeterminate, then STOP! 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 1 Evaluate 𝐥𝐢𝐦 𝒙→𝟐 𝒙 𝟐 −𝟒 𝒙−𝟐 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 2 Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝒙+𝟔 𝒙+𝟐 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 2 Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝒙+𝟔 𝒙+𝟐 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 3 Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝒆 𝟑𝒙 −𝟏 𝒙 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 4 Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝟏−𝐜𝐨𝐬 𝒙 𝒙 𝟐 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 4 Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝟏−𝐜𝐨𝐬 𝒙 𝒙 𝟐 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Your Turn Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝒙 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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Determinate vs. Indeterminate form
L’Hopital’s rule can be used to evaluate other indeterminate forms besides 0/0. (Make sure to change it to fractions FIRST) ± ∞ ∞ ∞ · 0 ∞ – ∞ 𝟏 ∞ 00 ∞ 𝟎 Determinate forms ∞ + ∞ = +∞ (Limit is to positive infinity) –∞ – ∞ = – ∞ (Limit is to negative infinity) 𝟎 ∞ = 0 (Limit is ZERO) 𝟎 −∞ = 0 (Limit is to positive infinity) These forms are NOT numbers. They are descriptions of function behaviors. Smaller Exponent/Bigger Exponent: (S/B) = 0 Bigger Exponent/Smaller Exponent: (B/S) = ±∞ 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 5 Evaluate 𝐥𝐢𝐦 𝒙→−∞ 𝒙 𝟐 𝒆 −𝒙 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 5 Evaluate 𝐥𝐢𝐦 𝒙→−∞ 𝒙 𝟐 𝒆 −𝒙 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 5 Evaluate 𝐥𝐢𝐦 𝒙→−∞ 𝒙 𝟐 𝒆 −𝒙 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 6 Evaluate 𝐥𝐢𝐦 𝒙→∞ 𝒆 𝒙 −𝟏 𝒙 𝟑 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 6 Evaluate 𝐥𝐢𝐦 𝒙→∞ 𝒆 𝒙 −𝟏 𝒙 𝟑 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 6 Evaluate 𝐥𝐢𝐦 𝒙→∞ 𝒆 𝒙 −𝟏 𝒙 𝟑 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Your Turn Evaluate 𝐥𝐢𝐦 𝒙→∞ 𝒆 𝒙 𝐥𝐧 𝒙 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 7 Evaluate 𝐥𝐢𝐦 𝒙→∞ 𝟏+ 𝟏 𝒙 𝒙 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 7 Evaluate 𝐥𝐢𝐦 𝒙→∞ 𝟏+ 𝟏 𝒙 𝒙 using L’Hopital’s Rule 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Example 8 The functions, 𝒇 and 𝒈, are each twice differentiable. Selected values of 𝒇 and 𝒈 and their derivatives are shown in the table. What is 𝐥𝐢𝐦 𝒙→𝟐 𝒇 𝒙 𝒈 𝒙 ? 𝑥 𝟏 𝟐 𝟑 𝒇 𝒙 𝟎 𝟒 𝒈 𝒙 𝟔 𝒇 ′ 𝒙 𝒈 ′ 𝒙 −𝟐 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝐥𝐢𝐦 𝒙→ 𝝅 𝟐 𝟏−𝐬𝐢𝐧 𝒙 𝟏+𝐜𝐨𝐬𝟐𝒙 (A) 𝟏 𝟒 (B) 𝟏 𝟐 (C) − 𝟏 𝟐 (D) − 𝟏 𝟒 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝐥𝐢𝐦 𝒙→ 𝝅 𝟐 𝟏−𝐬𝐢𝐧 𝒙 𝟏+𝐜𝐨𝐬𝟐𝒙 Vocabulary Connections and Process Answer 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule
Assignment Page odd, odd, 51B 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule
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