#Indeterminate Forms and L'Hospital's Rule

#Table of Contents

Introduction to Indeterminate Forms
Evaluating Limits Using L'Hospital's Rule
Practice Questions
Glossary
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:

$\frac{0}{0}$
$\frac{\pm \infty}{\pm \infty}$

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 limit of a quotient $\frac{f(x)}{g(x)}$ when substitution returns one of the indeterminate forms $\frac{0}{0}$ or $\frac{\pm \infty}{\pm \infty}$ .

For such a quotient function, L'Hospital's rule states:

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

This means you can take the derivatives of the numerator and denominator and attempt to evaluate the limit again in that form.

#Steps to Evaluate a Limit Using L’Hospital’s Rule

Check for Indeterminate Form:
- Ensure that the limit of the quotient results in one of the indeterminate forms given above, e.g., $\frac{0}{0}$ or $\frac{\pm \infty}{\pm \infty}$ .
Find Derivatives:
- Find the derivatives of the numerator and denominator of the quotient.
Evaluate the Limit:
- Check whether the limit $\lim_{{x \to a}} \frac{f'(x)}{g'(x)}$ exists.
Apply L'Hospital's Rule:
- If the limit exists, then $\lim_{{x \to a}} \frac{f(x)}{g(x)} = \lim_{{x \to a}} \frac{f'(x)}{g'(x)}$ .
Repeat if Necessary:
- If the limit still gives an indeterminate form, you may repeat the process by considering higher-order derivatives.

Exam Tip

Before using L'Hospital's rule to evaluate a limit, confirm that using substitution gives an indeterminate form. Otherwise, L'Hospital's rule is not valid.

#Worked Examples

**Example 1:** Evaluate the limit

\lim\_{{x \to \infty}} \frac{5x + 17}{4 - 3x}

using L'Hospital's rule.

Solution:

Check for indeterminate form:
- Substitution gives $\frac{\infty}{-\infty}$ , which is an indeterminate form.
Find derivatives:
- Derivative of the numerator: $\frac{d}{dx}(5x + 17) = 5$
- Derivative of the denominator: $\frac{d}{dx}(4 - 3x) = -3$
Apply L'Hospital's Rule:
- $\lim_{{x \to \infty}} \frac{5x + 17}{4 - 3x} = \lim_{{x \to \infty}} \frac{5}{-3}$
Evaluate the limit:
- $\lim_{{x \to \infty}} \frac{5}{-3} = -\frac{5}{3}$

Answer: $\lim_{{x \to \infty}} \frac{5x + 17}{4 - 3x} = -\frac{5}{3}$

**Example 2:** Evaluate the limit

\lim\_{{x \to 0}} \frac{x^3}{-2x + \sin(2x)}

using L'Hospital's rule.

Solution:

Check for indeterminate form:
- Substitution gives $\frac{0}{0}$ , which is an indeterminate form.
Find derivatives:
- Derivative of the numerator: $\frac{d}{dx}(x^3) = 3x^2$
- Derivative of the denominator: $\frac{d}{dx}(-2x + \sin(2x)) = -2 + 2\cos(2x)$
Apply L'Hospital's Rule:
- $\lim_{{x \to 0}} \frac{3x^2}{-2 + 2\cos(2x)}$
Check for indeterminate form:
- Substitution still gives $\frac{0}{0}$ , so repeat the process.
Find second derivatives:
- Derivative of the numerator: $\frac{d}{dx}(3x^2) = 6x$
- Derivative of the denominator: $\frac{d}{dx}(-2 + 2\cos(2x)) = -4\sin(2x)$
Apply L'Hospital's Rule again:
- $\lim_{{x \to 0}} \frac{6x}{-4\sin(2x)}$
Check for indeterminate form:
- Substitution still gives $\frac{0}{0}$ , so repeat the process.
Find third derivatives:
- Derivative of the numerator: $\frac{d}{dx}(6x) = 6$
- Derivative of the denominator: $\frac{d}{dx}(-4\sin(2x)) = -8\cos(2x)$
Apply L'Hospital's Rule again:
- $\lim_{{x \to 0}} \frac{6}{-8\cos(2x)}$
Evaluate the limit:
- $\lim_{{x \to 0}} \frac{6}{-8\cos(0)} = \frac{6}{-8(1)} = -\frac{3}{4}$

Answer: $\lim_{{x \to 0}} \frac{x^3}{-2x + \sin(2x)} = -\frac{3}{4}$

#Practice Questions

Practice Question

Question 1: Evaluate the limit $\lim_{{x \to \infty}} \frac{2x^2 - 3x + 1}{x^2 + 4x - 5}$ using L'Hospital's rule.

Practice Question

Question 2: Evaluate the limit $\lim_{{x \to 0}} \frac{\sin(x)}{x}$ using L'Hospital's rule.

Practice Question

Question 3: Evaluate the limit $\lim_{{x \to 0}} \frac{\ln(1 + x)}{x}$ using L'Hospital's rule.

#Glossary

Indeterminate Form: An expression that does not have a well-defined limit without further analysis, such as $\frac{0}{0}$ or $\frac{\pm \infty}{\pm \infty}$ .
L'Hospital's Rule: A method for evaluating limits of indeterminate forms by differentiating the numerator and denominator.

#Summary and Key Takeaways

#Summary

Indeterminate forms are expressions that require further analysis to determine their limits.
L'Hospital's rule is a powerful tool in calculus for evaluating limits of indeterminate forms by taking derivatives of the numerator and denominator.
Steps to use L'Hospital's rule:
1. Confirm the indeterminate form.
2. Differentiate the numerator and denominator.
3. Evaluate the new limit.
4. Repeat if necessary.

#Key Takeaways

Always check for the indeterminate form before applying L'Hospital's rule.
L'Hospital's rule can be applied iteratively for higher-order derivatives if the limit remains indeterminate.
Understanding and practicing L'Hospital's rule can simplify evaluating complex limits.

By following these guidelines and practicing the provided questions, you will strengthen your understanding of indeterminate forms and L'Hospital's rule, enhancing your exam performance.

#Indeterminate Forms and L'Hospital's Rule

#Table of Contents

Introduction to Indeterminate Forms
Evaluating Limits Using L'Hospital's Rule
Practice Questions
Glossary
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:

$\frac{0}{0}$
$\frac{\pm \infty}{\pm \infty}$

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

#Evaluating Limits Using L'Hospital's Rule

#What is L'Hospital’s Rule?

Key Concept

For such a quotient function, L'Hospital's rule states:

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

This means you can take the derivatives of the numerator and denominator and attempt to evaluate the limit again in that form.

#Steps to Evaluate a Limit Using L’Hospital’s Rule

Check for Indeterminate Form:
- Ensure that the limit of the quotient results in one of the indeterminate forms given above, e.g., $\frac{0}{0}$ or $\frac{\pm \infty}{\pm \infty}$ .
Find Derivatives:
- Find the derivatives of the numerator and denominator of the quotient.
Evaluate the Limit:
- Check whether the limit $\lim_{{x \to a}} \frac{f'(x)}{g'(x)}$ exists.
Apply L'Hospital's Rule:
- If the limit exists, then $\lim_{{x \to a}} \frac{f(x)}{g(x)} = \lim_{{x \to a}} \frac{f'(x)}{g'(x)}$ .
Repeat if Necessary:
- If the limit still gives an indeterminate form, you may repeat the process by considering higher-order derivatives.

Exam Tip

Before using L'Hospital's rule to evaluate a limit, confirm that using substitution gives an indeterminate form. Otherwise, L'Hospital's rule is not valid.

#Worked Examples

**Example 1:** Evaluate the limit

\lim\_{{x \to \infty}} \frac{5x + 17}{4 - 3x}

using L'Hospital's rule.

Solution:

Check for indeterminate form:
- Substitution gives $\frac{\infty}{-\infty}$ , which is an indeterminate form.
Find derivatives:
- Derivative of the numerator: $\frac{d}{dx}(5x + 17) = 5$
- Derivative of the denominator: $\frac{d}{dx}(4 - 3x) = -3$
Apply L'Hospital's Rule:
- $\lim_{{x \to \infty}} \frac{5x + 17}{4 - 3x} = \lim_{{x \to \infty}} \frac{5}{-3}$
Evaluate the limit:
- $\lim_{{x \to \infty}} \frac{5}{-3} = -\frac{5}{3}$

Answer: $\lim_{{x \to \infty}} \frac{5x + 17}{4 - 3x} = -\frac{5}{3}$

**Example 2:** Evaluate the limit

\lim\_{{x \to 0}} \frac{x^3}{-2x + \sin(2x)}

using L'Hospital's rule.

Solution:

Check for indeterminate form:
- Substitution gives $\frac{0}{0}$ , which is an indeterminate form.
Find derivatives:
- Derivative of the numerator: $\frac{d}{dx}(x^3) = 3x^2$
- Derivative of the denominator: $\frac{d}{dx}(-2x + \sin(2x)) = -2 + 2\cos(2x)$
Apply L'Hospital's Rule:
- $\lim_{{x \to 0}} \frac{3x^2}{-2 + 2\cos(2x)}$
Check for indeterminate form:
- Substitution still gives $\frac{0}{0}$ , so repeat the process.
Find second derivatives:
- Derivative of the numerator: $\frac{d}{dx}(3x^2) = 6x$
- Derivative of the denominator: $\frac{d}{dx}(-2 + 2\cos(2x)) = -4\sin(2x)$
Apply L'Hospital's Rule again:
- $\lim_{{x \to 0}} \frac{6x}{-4\sin(2x)}$
Check for indeterminate form:
- Substitution still gives $\frac{0}{0}$ , so repeat the process.
Find third derivatives:
- Derivative of the numerator: $\frac{d}{dx}(6x) = 6$
- Derivative of the denominator: $\frac{d}{dx}(-4\sin(2x)) = -8\cos(2x)$
Apply L'Hospital's Rule again:
- $\lim_{{x \to 0}} \frac{6}{-8\cos(2x)}$
Evaluate the limit:
- $\lim_{{x \to 0}} \frac{6}{-8\cos(0)} = \frac{6}{-8(1)} = -\frac{3}{4}$

Answer: $\lim_{{x \to 0}} \frac{x^3}{-2x + \sin(2x)} = -\frac{3}{4}$

#Practice Questions

Practice Question

Question 1: Evaluate the limit $\lim_{{x \to \infty}} \frac{2x^2 - 3x + 1}{x^2 + 4x - 5}$ using L'Hospital's rule.

Practice Question

Question 2: Evaluate the limit $\lim_{{x \to 0}} \frac{\sin(x)}{x}$ using L'Hospital's rule.

Practice Question

Question 3: Evaluate the limit $\lim_{{x \to 0}} \frac{\ln(1 + x)}{x}$ using L'Hospital's rule.

#Glossary

Indeterminate Form: An expression that does not have a well-defined limit without further analysis, such as $\frac{0}{0}$ or $\frac{\pm \infty}{\pm \infty}$ .
L'Hospital's Rule: A method for evaluating limits of indeterminate forms by differentiating the numerator and denominator.

#Summary and Key Takeaways

#Summary

Indeterminate forms are expressions that require further analysis to determine their limits.
L'Hospital's rule is a powerful tool in calculus for evaluating limits of indeterminate forms by taking derivatives of the numerator and denominator.
Steps to use L'Hospital's rule:
1. Confirm the indeterminate form.
2. Differentiate the numerator and denominator.
3. Evaluate the new limit.
4. Repeat if necessary.

#Key Takeaways

Always check for the indeterminate form before applying L'Hospital's rule.
L'Hospital's rule can be applied iteratively for higher-order derivatives if the limit remains indeterminate.
Understanding and practicing L'Hospital's rule can simplify evaluating complex limits.

By following these guidelines and practicing the provided questions, you will strengthen your understanding of indeterminate forms and L'Hospital's rule, enhancing your exam performance.

L'Hospital's Rule

Sarah Miller

#Indeterminate Forms and L'Hospital's Rule

#Table of Contents

#Introduction to Indeterminate Forms

#What is an Indeterminate Form?

#Evaluating Limits Using L'Hospital's Rule

#What is L'Hospital’s Rule?

#Steps to Evaluate a Limit Using L’Hospital’s Rule

#Worked Examples

#Practice Questions

#Glossary

#Summary and Key Takeaways

#Summary

#Key Takeaways

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L'Hospital's Rule

Sarah Miller

#Indeterminate Forms and L'Hospital's Rule

#Table of Contents

#Introduction to Indeterminate Forms

#What is an Indeterminate Form?

#Evaluating Limits Using L'Hospital's Rule

#What is L'Hospital’s Rule?

#Steps to Evaluate a Limit Using L’Hospital’s Rule

#Worked Examples

#Practice Questions

#Glossary

#Summary and Key Takeaways

#Summary

#Key Takeaways

Explore more resources

How are we doing?