Indeterminate Forms and L’Hopital’s Rule Section 8.7 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule

§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

§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

§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

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

§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

§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

§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

§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

§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

§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

§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

§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

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

§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

§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

§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

§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

§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

§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

§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

§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

§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

§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

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

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

§8.7: Indeterminate Forms and L’Hopital’s Rule Assignment Page 564 11-27 odd, 33-39 odd, 51B 11/27/2018 9:38 PM §8.7: Indeterminate Forms and L’Hopital’s Rule