Example of Using L'Hopital's Rule Twice to Evaluate a Limit

turksvids
19 Oct 201804:50
EducationalLearning
32 Likes 10 Comments

TLDRThis educational video guides viewers through the application of L'Hôpital's Rule in calculus to solve a problem where the limit yields an 'infinity over infinity' form. The instructor demonstrates using L'Hôpital's Rule multiple times, each step following the finding of 0/0 or infinity/infinity upon taking limits. The example involves taking derivatives of the numerator and denominator, employing chain and product rules for simplification. The video emphasizes both the mathematical process and proper notation for formal academic settings, illustrating how to persistently simplify and solve challenging limit problems step-by-step.

Takeaways
  • 📚 L'Hôpital's Rule is used to evaluate limits that result in the indeterminate forms of 0/0 or ∞/∞.
  • 🔍 When applying L'Hôpital's Rule, take the derivative of the numerator and the denominator separately.
  • 🔑 It's common to apply L'Hôpital's Rule more than once to resolve the limit, as the new form might still be indeterminate.
  • 📈 The script demonstrates the process of applying L'Hôpital's Rule twice to evaluate the limit of a specific function as x approaches infinity.
  • 🧮 The first application of L'Hôpital's Rule involves using the chain rule for the derivative of the natural log of x squared over e to the 2x.
  • 📉 After the first application, the limit still results in an indeterminate form, prompting a second application of L'Hôpital's Rule.
  • 🔄 The second derivative requires the use of both the chain rule and the product rule to simplify the expression.
  • 🎯 Simplification is crucial after each application of L'Hôpital's Rule to make the algebraic process easier.
  • 📝 The script emphasizes the importance of showing formal work, especially in an exam or academic setting.
  • 💡 It's highlighted that while the left-hand side work is more common in homework, the right-hand side represents a more formal approach.
  • ✅ The final answer is obtained after successive applications of L'Hôpital's Rule, resulting in a determinate limit.
  • 📋 The presenter also provides tips on how to present work neatly, which is helpful for understanding and grading purposes.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is the application of L'Hôpital's rule in evaluating limits, specifically when the limit results in an indeterminate form such as 0/0 or ∞/∞.

  • What is L'Hôpital's rule used for?

    -L'Hôpital's rule is used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞ by taking the derivatives of the numerator and denominator.

  • What is the first step when applying L'Hôpital's rule?

    -The first step is to take the derivatives of the numerator and denominator and then re-evaluate the limit with the new expressions.

  • Why might one need to apply L'Hôpital's rule more than once?

    -One might need to apply L'Hôpital's rule more than once if after the first application, the new limit is still in an indeterminate form such as 0/0 or ∞/∞.

  • What is the example limit given in the video?

    -The example limit is as X approaches infinity of the quantity natural log of X squared over e to the 2x.

  • What does the video suggest about the initial evaluation of the given limit?

    -The video suggests that initially, the limit appears to be ∞/∞ because both the numerator (natural log of X squared) and the denominator (e to the 2x) tend to infinity as X approaches infinity.

  • What is the derivative of natural log of X?

    -The derivative of natural log of X (ln X) is 1/X.

  • What is the derivative of e to the 2x?

    -The derivative of e to the 2x with respect to X is 2 * e to the 2x, using the chain rule for differentiation.

  • What is the chain rule in calculus?

    -The chain rule is a method in calculus for finding the derivative of a composite function, which is a function composed of two or more functions.

  • What is the product rule in calculus?

    -The product rule is a formula used in calculus to find the derivative of a product of two functions, stating that the derivative of the product is the derivative of the first function times the second function plus the first function times the derivative of the second function.

  • Why is it important to simplify after applying L'Hôpital's rule?

    -Simplifying after applying L'Hôpital's rule makes the subsequent steps in the limit evaluation process easier and can often reveal the limit's value more clearly.

  • What is the final outcome of applying L'Hôpital's rule multiple times in the given example?

    -In the given example, after applying L'Hôpital's rule multiple times, the limit simplifies to a point where it can be determined, and one does not need to apply the rule further.

  • What does the video suggest for presenting a formal solution to the limit problem?

    -The video suggests that a formal solution should include clear steps, the application of L'Hôpital's rule, and a final evaluation of the limit after simplification.

Outlines
00:00
📚 Introduction to L'Hôpital's Rule Application

The video begins with an introduction to L'Hôpital's rule, which is utilized when taking limits results in an indeterminate form such as 0/0 or ∞/∞. The presenter illustrates the process with an example involving the limit as X approaches infinity of the natural logarithm of X squared over e to the power of 2X. Initially, it's determined that both the numerator and denominator tend to infinity, hence applying L'Hôpital's rule is necessary. The presenter demonstrates both a quick approach and a more formal method, emphasizing the importance of showing work, especially in academic or formal settings.

Mindmap
Keywords
💡L'Hôpital's Rule
L'Hôpital's Rule is a mathematical method used to evaluate limits of functions that result in an indeterminate form, such as 0/0 or ∞/∞. In the video, it is applied multiple times to find the limit of the natural logarithm of x squared over e to the 2x as x approaches infinity. The rule is central to the video's theme of solving complex limit problems.
💡Limit
A limit in calculus is the value that a function approaches as the input (in this case, x) approaches a certain value. The video discusses finding the limit as x approaches infinity, which is a common scenario in calculus. The concept of limits is fundamental to understanding the theme of the video.
💡Natural Logarithm
The natural logarithm, often denoted as ln(x), is the logarithm to the base e. In the video, the natural logarithm of x squared is part of the function whose limit is being evaluated. It is used to demonstrate how the value of the function component tends to infinity as x approaches infinity.
💡e (Euler's Number)
e, known as Euler's number, is an important mathematical constant approximately equal to 2.71828. In the video, e is used as the base of the exponential function e^(2x), which also tends to infinity as x approaches infinity. It's a key component in the function under the limit being evaluated.
💡Indeterminate Form
An indeterminate form, such as 0/0 or ∞/∞, occurs when the limit of a function is not immediately clear and requires further manipulation to resolve. The video begins with an indeterminate form of ∞/∞, which prompts the use of L'Hôpital's Rule.
💡Derivative
The derivative of a function represents the rate at which the function is changing at a given point. In applying L'Hôpital's Rule, derivatives of the numerator and denominator are calculated. For example, the derivative of ln(x) is 1/x, which is used in the video to simplify the expression after applying the rule.
💡Chain Rule
The chain rule is a fundamental theorem used for finding the derivatives of composite functions. In the video, it is used to find the derivative of e^(2x), which is part of the denominator after applying L'Hôpital's Rule. The chain rule is essential for simplifying the expression to a form that can be evaluated.
💡Product Rule
The product rule is a formula used to find the derivative of a product of two functions. In the video, it is mentioned in the context of differentiating a product in the denominator after applying L'Hôpital's Rule for the second time. The product rule is key to further simplifying the expression.
💡Infinity
Infinity is a concept that refers to an unbounded quantity that is larger than any number. In the video, both the numerator and the denominator of the original function tend to infinity as x approaches infinity, leading to the indeterminate form that initiates the application of L'Hôpital's Rule.
💡Simplify
Simplifying a mathematical expression means to rewrite it in a more straightforward form that is easier to understand or manipulate. The video emphasizes the importance of simplifying after each application of L'Hôpital's Rule to make the subsequent steps more manageable.
💡Formal Work
Formal work refers to a detailed and structured approach to solving a problem, which is often required in academic or professional settings. In the video, the presenter contrasts the formal work with the more common approach, highlighting the importance of thorough explanations and clear steps, especially when showing work in an exam or formal assignment.
Highlights

Introduction to using l'Hôpital's rule for limits that result in 0/0 or ∞/∞.

The example involves the limit as X approaches infinity of the natural log of X squared over e to the 2x.

Initial assessment suggests both the numerator and denominator tend towards infinity.

Application of l'Hôpital's rule by taking the derivative of the numerator and denominator.

Use of the chain rule for the derivative of the natural log of X squared.

Simplification of the expression after applying l'Hôpital's rule.

Re-evaluation of the limit after applying l'Hôpital's rule, which still results in ∞/∞.

Applying l'Hôpital's rule a second time to the new limit.

Using the product rule for the derivative of the denominator, which is a product of two functions.

Further algebraic simplification of the expression after the second application of l'Hôpital's rule.

Observation that as X approaches infinity, the new expression simplifies to 0.

The final answer is obtained after the second application of l'Hôpital's rule.

Demonstration of how to present formal work for a free-response question, like on the AP exam.

Comparison between the work done on the left-hand side (more common in homework) and the right-hand side (formal work).

Emphasis on the importance of simplifying expressions after each application of l'Hôpital's rule.

The process may need to be repeated if the new limit is still ∞/∞.

Guidance on how to present work neatly and clearly for better understanding.

The video concludes with a summary of the process and an encouragement for further practice.

Transcripts

Browse More Related Video

Lesson 5 - L'hopitals Rule (Calculus 2 Tutor)

Calculus Chapter 1 Lecture 8 l'Hopital's Rule

Using L'Hopital's Rule to show that exponentials dominate polynomials

Limits at infinity of quotients with square roots (even power) | AP Calculus AB | Khan Academy

Ryan and Nick do L'hopital's rule

Limits and Continuous Functions

Rate This

5.0 / 5 (0 votes)

Thanks for rating:

Related Tags
L'Hôpital's RuleCalculusLimitsInfinityDerivativesChain RuleProduct RuleMathematicsEducationSolving LimitsAP Exam