1. 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

2. §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

3. §8.7: Indeterminate Forms and L’Hopital’s Rule
Review
Evaluate 𝐥𝐢𝐦 𝒙→𝟐 𝒙 𝟐 −𝟒 𝒙−𝟐 using the limit process
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§8.7: Indeterminate Forms and L’Hopital’s Rule

4. §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

5. 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).
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§8.7: Indeterminate Forms and L’Hopital’s Rule

6. §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!
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§8.7: Indeterminate Forms and L’Hopital’s Rule

7. §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
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§8.7: Indeterminate Forms and L’Hopital’s Rule

8. §8.7: Indeterminate Forms and L’Hopital’s Rule
Example 2
Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝒙+𝟔 𝒙+𝟐 using L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule

9. §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

10. §8.7: Indeterminate Forms and L’Hopital’s Rule
Example 3
Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝒆 𝟑𝒙 −𝟏 𝒙 using L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule

11. §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

12. §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

13. §8.7: Indeterminate Forms and L’Hopital’s Rule
Your Turn
Evaluate 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝟐𝒙 𝒙 using L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule

14. 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) = ±∞
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§8.7: Indeterminate Forms and L’Hopital’s Rule

15. §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

16. §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

17. §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

18. §8.7: Indeterminate Forms and L’Hopital’s Rule
Example 6
Evaluate 𝐥𝐢𝐦 𝒙→∞ 𝒆 𝒙 −𝟏 𝒙 𝟑 using L’Hopital’s Rule
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§8.7: Indeterminate Forms and L’Hopital’s Rule

19. §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

20. §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

21. §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

22. §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

23. §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

24. §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
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§8.7: Indeterminate Forms and L’Hopital’s Rule

25. 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
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§8.7: Indeterminate Forms and L’Hopital’s Rule

26. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝐥𝐢𝐦 𝒙→ 𝝅 𝟐 𝟏−𝐬𝐢𝐧 𝒙 𝟏+𝐜𝐨𝐬𝟐𝒙
Vocabulary
Connections and Process
Answer
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§8.7: Indeterminate Forms and L’Hopital’s Rule

27. §8.7: Indeterminate Forms and L’Hopital’s Rule
Assignment
Page odd, odd, 51B
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§8.7: Indeterminate Forms and L’Hopital’s Rule