Using L'Hopitals Rule for Determining Limits in Indeterminate Forms

Benjamin Wright
7 min read
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Study Guide Overview
This study guide covers L'Hôpital's Rule for evaluating indeterminate forms of limits (0/0 or ±∞/∞). It explains the rule, provides a step-by-step example, and offers practice problems with solutions. The guide emphasizes verifying conditions for applying the rule, especially for FRQs. Key takeaways for the exam include recognizing indeterminate forms, applying derivative rules correctly, and showing all necessary steps. Common mistakes and time management tips are also addressed.
#AP Calculus AB/BC: L'Hôpital's Rule - Your Ultimate Guide 🚀
Hey there, future AP Calculus master! Let's dive into L'Hôpital's Rule, a super handy tool for tackling those tricky indeterminate limits. This guide is designed to be your go-to resource, especially the night before the exam. Let's make sure you're feeling confident and ready! 💪
#4.7: L'Hôpital's Rule for Indeterminate Forms
Remember those limits that gave us or ? Those are called indeterminate forms, and they're where L'Hôpital's Rule shines. Instead of algebraic manipulations, we get to use derivatives! 🥳
#📏 What is L'Hôpital's Rule?
L'Hôpital's Rule states that if results in or , then:
In simple terms, take the derivative of the top and bottom separately, then try the limit again! 💡
Important Note: This is NOT the quotient rule! L'Hôpital's Rule is ONLY for indeterminate limits. Don't use it anywhere else! 🙅♀️
#✏️ L'Hôpital's Rule: Step-by-Step
Let's walk through an example together. Evaluate:
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Check for Indeterminate Form: Plugging in gives us . ✅
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Verify Conditions (FRQ Must-Do):
Since both limits are 0, we can apply L'Hôpital's Rule. (Make sure to state this in the exam!)
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Apply L'Hôpital's Rule:
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Take Derivatives:
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Evaluate the Limit:
So, . 🎉
Always show that the limit is in indeterminate form before applying L'Hôpital's Rule in FRQs. This will get you points! 📝
#📝 Practice Time!
Ready to try it yourself? Here are a couple of practice problems. Treat them like FRQs and show all your steps!
#❓ L'Hôpital's Rule: Practice Problems
#Question 1:
#Question 2:
#✅ L'Hôpital's Rule: Solutions
#Question 1:
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Indeterminate Form: Plugging in gives .
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Verify Conditions:
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Apply L'Hôpital's Rule:
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Take Derivatives:
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Evaluate the Limit:
Thus, .
#Question 2:
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Indeterminate Form: Plugging in gives .
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Verify Conditions:
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Apply L'Hôpital's Rule:
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Take Derivatives:
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Simplify and Evaluate:
Therefore, .
#💫 Final Exam Focus
Key Takeaways for Exam Day:
- Indeterminate Forms: Always check for or before using L'Hôpital's Rule.
- FRQ Steps: Show the limits of the numerator and denominator separately. State that you're applying L'Hôpital's Rule.
- Derivatives: Make sure you know your derivative rules inside and out! (Trig, polynomials, exponentials, etc.)
- Simplification: Simplify after taking derivatives. Sometimes you might need to apply L'Hôpital's Rule multiple times.
- Connections: L'Hôpital's Rule often appears in combination with other concepts, so be ready to use it in different contexts.
Time Management: If you get stuck, move on and come back. Don't spend too much time on one question. Remember, partial credit is your friend!
Common Pitfalls:
- Forgetting to check for indeterminate form first.
- Applying L'Hôpital's Rule when it's not needed.
- Incorrectly applying derivative rules.
- Not showing the required steps in FRQs.
Quick Tip: L'Hôpital's Rule can be a lifesaver on the exam, but remember to use it wisely! It's not a magic wand for all limits. ✨
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Practice Question
Practice Questions
#Multiple Choice Questions
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Evaluate : (A) 0 (B) 1 (C) 3 (D) Does not exist
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Evaluate : (A) 0 (B) 2/3 (C) 5/1 (D) Does not exist
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Evaluate : (A) 0 (B) 1 (C) -1 (D) Does not exist
#Free Response Question
Consider the function .
(a) Show that is an indeterminate form. (2 points)
(b) Evaluate . (4 points)
(c) Let . Evaluate . (3 points)
#FRQ Scoring Rubric
(a)
- 1 point: Shows
- 1 point: Shows
(b)
- 1 point: Applies L'Hôpital's Rule once correctly
- 1 point: Takes the derivatives correctly
- 1 point: Applies L'Hôpital's Rule again correctly
- 1 point: Evaluates the limit correctly
(c)
- 1 point: Shows and
- 1 point: Applies L'Hôpital's Rule correctly
- 1 point: Evaluates the limit correctly
#Solutions to Practice Questions
#Multiple Choice
- (C) 3 (Apply L'Hôpital's Rule)
- (B) 2/3 (Apply L'Hôpital's Rule or compare leading coefficients)
- (B) 1 (Apply L'Hôpital's Rule)
#Free Response Question
(a)
Since both limits are 0, the limit is of the form , which is an indeterminate form.
(b)
Applying L'Hôpital's Rule once:
This is still an indeterminate form , so apply L'Hôpital's Rule again:
Thus, .
(c)
Applying L'Hôpital's Rule:
Thus, .
You've got this! Keep practicing, stay calm, and remember all the cool tricks you've learned. You're going to rock this exam! 🌟
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