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Using L'Hopitals Rule for Determining Limits in Indeterminate Forms

Benjamin Wright

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 00\frac{0}{0} or ±\pm\frac{\infty}{\infty}? 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?

Key Concept

L'Hôpital's Rule states that if limxaf(x)g(x)\lim_{x\to a}\frac{f(x)}{g(x)} results in 00\frac{0}{0} or ±\pm\frac{\infty}{\infty}, then:

limxaf(x)g(x)=limxaf(x)g(x)\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}

In simple terms, take the derivative of the top and bottom separately, then try the limit again! 💡

Common Mistake

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:

limxπ2cos(x)xπ2\lim_{x \to \frac{\pi}{2}}\frac{\cos(x)}{x-\frac{\pi}{2}}

  1. Check for Indeterminate Form: Plugging in x=π2x = \frac{\pi}{2} gives us 00\frac{0}{0}. ✅

  2. Verify Conditions (FRQ Must-Do):

    limxπ2cos(x)=0\lim_{x \to \frac{\pi}{2}}\cos(x) = 0

    limxπ2xπ2=0\lim_{x \to \frac{\pi}{2}}x - \frac{\pi}{2} = 0

    Since both limits are 0, we can apply L'Hôpital's Rule. (Make sure to state this in the exam!)

  3. Apply L'Hôpital's Rule:

    limxπ2cos(x)xπ2=limxπ2ddx[cos(x)]ddx[xπ2]\lim_{x \to \frac{\pi}{2}}\frac{\cos(x)}{x-\frac{\pi}{2}} = \lim_{x\to \frac{\pi}{2}}\frac{\frac{d}{dx}[\cos(x)]}{\frac{d}{dx}[x-\frac{\pi}{2}]}

  4. Take Derivatives:

    limxπ2sin(x)1\lim_{x\to \frac{\pi}{2}}\frac{-\sin(x)}{1}

  5. Evaluate the Limit:

    =sin(π2)1=1= \frac{-\sin(\frac{\pi}{2})}{1} = -1

So, limxπ2cos(x)xπ2=1\lim_{x \to \frac{\pi}{2}}\frac{\cos(x)}{x-\frac{\pi}{2}} = -1. 🎉

Exam Tip

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:

limx0tan(x)7x+tan(x)\lim_{x\to 0}\frac{\tan(x)}{7x+\tan(x)}

Question 2:

limx3x287x2+21\lim_{x\to \infty}\frac{3x^2-8}{7x^2+21}

✅ L'Hôpital's Rule: Solutions

Question 1:

  1. Indeterminate Form: Plugging in x=0x=0 gives 00\frac{0}{0}.

  2. Verify Conditions:

    limx0tan(x)=0\lim_{x\to 0}\tan(x) = 0

    limx07x+tan(x)=0\lim_{x\to 0}7x + \tan(x) = 0

  3. Apply L'Hôpital's Rule:

    limx0tan(x)7x+tan(x)=limx0ddx[tan(x)]ddx[7x+tan(x)]\lim_{x\to 0}\frac{\tan(x)}{7x+\tan(x)} = \lim_{x\to 0}\frac{\frac{d}{dx}[\tan(x)]}{\frac{d}{dx}[7x+\tan(x)]}

  4. Take Derivatives:

    limx0sec2(x)7+sec2(x)\lim_{x\to 0}\frac{\sec^2(x)}{7+\sec^2(x)}

  5. Evaluate the Limit:

    =sec2(0)7+sec2(0)=17+1=18= \frac{\sec^2(0)}{7+\sec^2(0)} = \frac{1}{7+1} = \frac{1}{8}

    Thus, limx0tan(x)7x+tan(x)=18\lim_{x\to 0}\frac{\tan(x)}{7x+\tan(x)} = \frac{1}{8}.

Question 2:

  1. Indeterminate Form: Plugging in x=x = \infty gives \frac{\infty}{\infty}.

  2. Verify Conditions:

    limx3x28=\lim_{x\to \infty}3x^2 - 8 = \infty

    limx7x2+21=\lim_{x\to \infty}7x^2 + 21 = \infty

  3. Apply L'Hôpital's Rule:

    limx3x287x2+21=limxddx[3x28]ddx[7x2+21]\lim_{x\to \infty}\frac{3x^2-8}{7x^2+21} = \lim_{x\to \infty}\frac{\frac{d}{dx}[3x^2-8]}{\frac{d}{dx}[7x^2+21]}

  4. Take Derivatives:

    limx6x14x\lim_{x\to \infty}\frac{6x}{14x}

  5. Simplify and Evaluate:

    =614=37= \frac{6}{14} = \frac{3}{7}

    Therefore, limx3x287x2+21=37\lim_{x\to \infty}\frac{3x^2-8}{7x^2+21} = \frac{3}{7}.


💫 Final Exam Focus

Key Takeaways for Exam Day:

  • Indeterminate Forms: Always check for 00\frac{0}{0} or ±\pm\frac{\infty}{\infty} 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.
Exam Tip

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 Mistake

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 Fact

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. ✨

Practice Question

Practice Questions

Multiple Choice Questions

  1. Evaluate limx0sin(3x)x\lim_{x \to 0} \frac{\sin(3x)}{x}: (A) 0 (B) 1 (C) 3 (D) Does not exist

  2. Evaluate limx2x2+5x3x21\lim_{x \to \infty} \frac{2x^2 + 5x}{3x^2 - 1}: (A) 0 (B) 2/3 (C) 5/1 (D) Does not exist

  3. Evaluate limx1ln(x)x1\lim_{x \to 1} \frac{\ln(x)}{x-1}: (A) 0 (B) 1 (C) -1 (D) Does not exist

Free Response Question

Consider the function f(x)=ex1xx2f(x) = \frac{e^x - 1 - x}{x^2}.

(a) Show that limx0f(x)\lim_{x \to 0} f(x) is an indeterminate form. (2 points)

(b) Evaluate limx0f(x)\lim_{x \to 0} f(x). (4 points)

(c) Let g(x)=ex1xxg(x) = \frac{e^x - 1 - x}{x}. Evaluate limx0g(x)\lim_{x \to 0} g(x). (3 points)

FRQ Scoring Rubric

(a)

  • 1 point: Shows limx0(ex1x)=0\lim_{x \to 0} (e^x - 1 - x) = 0
  • 1 point: Shows limx0x2=0\lim_{x \to 0} x^2 = 0

(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 limx0(ex1x)=0\lim_{x \to 0} (e^x - 1 - x) = 0 and limx0x=0\lim_{x \to 0} x = 0
  • 1 point: Applies L'Hôpital's Rule correctly
  • 1 point: Evaluates the limit correctly

Solutions to Practice Questions

Multiple Choice

  1. (C) 3 (Apply L'Hôpital's Rule)
  2. (B) 2/3 (Apply L'Hôpital's Rule or compare leading coefficients)
  3. (B) 1 (Apply L'Hôpital's Rule)

Free Response Question

(a)

limx0(ex1x)=e010=11=0\lim_{x \to 0} (e^x - 1 - x) = e^0 - 1 - 0 = 1 - 1 = 0

limx0x2=02=0\lim_{x \to 0} x^2 = 0^2 = 0

Since both limits are 0, the limit is of the form 00\frac{0}{0}, which is an indeterminate form.

(b)

Applying L'Hôpital's Rule once:

limx0ex1xx2=limx0ex12x\lim_{x \to 0} \frac{e^x - 1 - x}{x^2} = \lim_{x \to 0} \frac{e^x - 1}{2x}

This is still an indeterminate form 00\frac{0}{0}, so apply L'Hôpital's Rule again:

limx0ex12x=limx0ex2=e02=12\lim_{x \to 0} \frac{e^x - 1}{2x} = \lim_{x \to 0} \frac{e^x}{2} = \frac{e^0}{2} = \frac{1}{2}

Thus, limx0f(x)=12\lim_{x \to 0} f(x) = \frac{1}{2}.

(c)

limx0(ex1x)=0\lim_{x \to 0} (e^x - 1 - x) = 0

limx0x=0\lim_{x \to 0} x = 0

Applying L'Hôpital's Rule:

limx0ex1xx=limx0ex11=e01=11=0\lim_{x \to 0} \frac{e^x - 1 - x}{x} = \lim_{x \to 0} \frac{e^x - 1}{1} = e^0 - 1 = 1 - 1 = 0

Thus, limx0g(x)=0\lim_{x \to 0} g(x) = 0.


You've got this! Keep practicing, stay calm, and remember all the cool tricks you've learned. You're going to rock this exam! 🌟

Question 1 of 8

Which of the following limits results in an indeterminate form that L'Hôpital's Rule can be applied to? 🤔

limx2x+1x1\lim_{x \to 2} \frac{x+1}{x-1}

limx0sin(x)x2+1\lim_{x \to 0} \frac{\sin(x)}{x^2 + 1}

limx0sin(x)x\lim_{x \to 0} \frac{\sin(x)}{x}

limx1x2x\lim_{x \to 1} \frac{x^2}{x}