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L'Hospital's Rule

Sarah Miller

Sarah Miller

6 min read

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Study Guide Overview

This study guide covers indeterminate forms and L'Hospital's Rule. It explains how to identify indeterminate forms like 0/0 and ∞/∞. The guide details L'Hospital's Rule, including the steps for applying it and working through example problems. It also provides practice questions and a glossary of key terms.

Indeterminate Forms and L'Hospital's Rule

Table of Contents

  1. Introduction to Indeterminate Forms
  2. Evaluating Limits Using L'Hospital's Rule
  3. Practice Questions
  4. Glossary
  5. Summary and Key Takeaways

Introduction to Indeterminate Forms

What is an Indeterminate Form?

Key Concept

An indeterminate form is a mathematical expression that does not have a well-defined limit without further analysis. The most common indeterminate forms are:

  • 00\frac{0}{0}
  • ±∞±∞\frac{\pm \infty}{\pm \infty}
  • The value of an indeterminate form is undefined.
    • Dividing by 0 always gives an undefined expression.
    • For instance, ∞∞\frac{\infty}{\infty} is not equal to 1. - ∞\infty is not a number, so it can't be canceled to simplify a fraction.
  • Sometimes, attempting to evaluate a limit using substitution leads to one of the indeterminate forms given above.
    • L'Hospital's rule provides a method for dealing with limits of that form.
Exam Tip

Other limit methods will sometimes work when substitution gives an indeterminate form. These include algebraic simplification, multiplying by conjugates, or multiplying by reciprocals. Refer to the 'Evaluating Limits Analytically' study guide for more information.

Evaluating Limits Using L'Hospital's Rule

What is L'Hospital’s Rule?

Key Concept

L'Hospital's rule (sometimes written as L’Hôpital’s rule) is a method for finding the value of certain limits using calculus. Specifically, it allows us to attempt to evaluate the lim...