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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 lim⁡x→af(x)g(x)\lim_{x\to a}\frac{f(x)}{g(x)} results in 00\frac{0}{0} or ±∞∞\pm\frac{\infty}{\infty}, then:

lim⁡x→af(x)g(x)=lim⁡x→af′(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:

lim⁡x→π2cos⁡(x)x−π2\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):

    lim⁡x→π2cos⁡(x)=0\lim_{x \to \frac{\pi}{2}}\cos(x) = 0

    lim⁡x→π2x−π2=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:

Previous Topic - Approximating Values of a Function Using Local Linearity and LinearizationNext Topic - Analytical Applications of Differentiation

Question 1 of 8

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

lim⁡x→2x+1x−1\lim_{x \to 2} \frac{x+1}{x-1}

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

lim⁡x→0sin⁡(x)x\lim_{x \to 0} \frac{\sin(x)}{x}

lim⁡x→1x2x\lim_{x \to 1} \frac{x^2}{x}