L'Hôpital's rule example 3 | Derivative applications | Differential Calculus | Khan Academy

Khan Academy
7 Jun 201007:50
EducationalLearning
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TLDRThe video script presents a step-by-step mathematical approach to finding the limit of a function as x approaches 1. Initially, plugging in the value of 1 results in an undefined form, prompting the use of algebraic manipulation to create an l'Hopital's rule scenario. Through a series of derivative calculations, the limit is eventually determined to be 1/2, showcasing the power of l'Hopital's rule in resolving complex limits.

Takeaways
  • 💡 The initial expression evaluated at x=1 results in an undefined form, not suitable for L'Hopital's rule.
  • 💥 By algebraically adding the two expressions, a common denominator is formed, enabling further simplification.
  • 💬 When the expressions are added and simplified, it results in an indeterminate form 0/0, making L'Hopital's rule applicable.
  • 🔮 Applying L'Hopital's rule involves taking derivatives of the numerator and denominator and evaluating the limit again.
  • 📚 The product rule is used for differentiation when the numerator's terms are products of functions.
  • 📈 Simplification after differentiation is crucial for easier evaluation of the limit.
  • 📌 The limit evaluation post-simplification might still result in an indeterminate form, necessitating the reapplication of L'Hopital's rule.
  • 📝 Continual application of L'Hopital's rule and simplification leads to resolving the indeterminate form.
  • 📡 The final evaluation after repeated application of L'Hopital's rule yields a concrete limit value.
  • 📖 The process demonstrates the power of algebraic manipulation and L'Hopital's rule in solving complex limit problems.
Q & A

  • What is the initial expression the transcript is trying to evaluate at x approaches 1?

    -The initial expression is the limit as x approaches 1 of x/(x-1) - 1/natural log(x).

  • Why is the direct substitution of x=1 not possible for the given expression?

    -Direct substitution of x=1 is not possible because it results in an undefined form of 1/0 - 1/0, which is not an indeterminate form that can be resolved using l'Hopital's rule.

  • What is the significance of the natural log of 1 in the evaluation of the expression?

    -The natural log of 1 is significant because it equals 0. This simplifies the expression when x=1 is substituted, as all terms involving natural log(1) become 0.

  • How does the transcript suggest we can manipulate the original expression to make it suitable for l'Hopital's rule?

    -The transcript suggests that by adding the original expression to itself, we can create a common denominator and then simplify to get an expression that, when x approaches 1, results in an indeterminate form of 0/0 suitable for l'Hopital's rule.

  • What is the first step in applying l'Hopital's rule to the manipulated expression?

    -The first step is to find the derivatives of the numerator and the denominator. The limit as x approaches 1 of the expression is then equal to the limit of the ratio of these derivatives.

  • What is the result of the first application of l'Hopital's rule?

    -The first application of l'Hopital's rule results in a new indeterminate form of 0/0, which requires a second application of the rule to resolve.

  • How does the second application of l'Hopital's rule lead to a determinate result?

    -The second application of l'Hopital's rule simplifies the expression to a form where the numerator evaluates to 1 and the denominator evaluates to 1/2 when x approaches 1, leading to a determinate result of 1/2.

  • What is the final answer for the limit as x approaches 1 of the original expression?

    -The final answer for the limit as x approaches 1 of the original expression is 1/2.

  • What mathematical concept is used to resolve the undefined form 1/0 - 1/0?

    -The mathematical concept used to resolve the undefined form 1/0 - 1/0 is l'Hopital's rule, which allows for the evaluation of limits of indeterminate forms by taking the derivatives of the numerator and the denominator.

  • What is the role of algebraic manipulation in solving this problem?

    -Algebraic manipulation plays a crucial role in this problem by transforming the original expression into a form that can be analyzed using l'Hopital's rule, which would not have been possible without the manipulation.

Outlines
00:00
📚 Limit Calculation and L'Hopital's Rule

The paragraph discusses the process of finding the limit of a given expression as x approaches 1. Initially, plugging in the value of x results in an undefined form, which does not fit the criteria for L'Hopital's rule. However, by algebraically manipulating the expression and combining terms, the speaker creates a form that does fit the rule's criteria. The speaker then applies L'Hopital's rule by taking the derivatives of the numerator and the denominator and evaluates the limit, resulting in an indeterminate form. The process is repeated until a final answer is reached.

05:04
🔢 Application of L'Hopital's Rule for Indeterminate Forms

This paragraph continues the discussion on limit calculation, focusing on the application of L'Hopital's rule for indeterminate forms. The speaker evaluates the limit as x approaches 1 for a transformed expression, which initially results in another indeterminate form. By applying L'Hopital's rule again, taking the derivatives of both the numerator and the denominator, the speaker simplifies the expression to a determinate form. The final calculation yields a result of 1/2, demonstrating the effectiveness of L'Hopital's rule in resolving complex limit problems.

Mindmap
Keywords
💡limit
In the context of the video, 'limit' refers to a fundamental concept in calculus that describes the behavior of a function when the input (or argument) approaches a particular value. The process of finding the limit of a function as x approaches 1 is central to the problem-solving narrative of the video. The term is used to analyze the behavior of the given expression as x gets arbitrarily close to 1, which is essential for determining the value or behavior of the function at that point.
💡undefined
The term 'undefined' in the video script refers to a situation where a mathematical expression does not have a meaningful value. Specifically, when the video discusses plugging in 1 into the expression, it initially results in an undefined form, 1/0 - 1/0, which is not a number. This highlights a challenge in evaluating the expression directly and indicates the need for further mathematical manipulation or the application of rules like l'Hopital's rule to find the limit.
💡l'Hopital's rule
L'Hopital's rule is a method in calculus that helps find the limit of a function when it appears to be undefined or indeterminate, such as 0/0 or ∞/∞, by taking the derivatives of the numerator and denominator and then re-evaluating the limit. In the video, this rule is crucial for resolving the indeterminate form that arises when trying to find the limit of the given expression. The application of l'Hopital's rule is demonstrated through a step-by-step process, which eventually leads to a determinate result.
💡natural log
The 'natural log' or 'ln' is a mathematical function that calculates the logarithm of a number to the base 'e', where 'e' is the mathematical constant approximately equal to 2.71828. In the video, the natural log is a part of the given expression, and understanding its properties, such as ln(1) being 0, is critical in the process of finding the limit. The natural log function is used in the expression that the video aims to evaluate and is manipulated algebraically to facilitate the application of l'Hopital's rule.
💡algebraic manipulation
Algebraic manipulation refers to the process of transforming and rearranging mathematical expressions using the rules of algebra to achieve a desired form or to simplify the expression. In the video, algebraic manipulation is key to resolving the initial undefined form of the expression by adding the expression to itself, which allows for the application of l'Hopital's rule and ultimately finding the limit as x approaches 1.
💡indeterminate form
An 'indeterminate form' is a type of mathematical expression that does not have a clear value, often because it involves division by zero or operations that result in infinity. In the context of the video, the initial attempt to evaluate the expression at x=1 results in an indeterminate form, which cannot be directly interpreted. However, this form is not the 0/0 type that l'Hopital's rule can resolve, leading to the need for additional algebraic steps before applying the rule.
💡product rule
The 'product rule' is a fundamental rule in calculus used for finding the derivative of a product of two functions. It states that the derivative of a product is the derivative of the first function times the second function plus the first function times the derivative of the second function. In the video, the product rule is applied when differentiating the numerator and denominator during the application of l'Hopital's rule, which is crucial for simplifying the expression and finding the limit.
💡derivative
The 'derivative' is a core concept in calculus that represents the rate of change or the slope of a function at a particular point. It is used to analyze the behavior of a function, such as its increasing or decreasing nature, and to find tangent lines or maximize/minimize values. In the video, derivatives are calculated for both the numerator and the denominator of the expression to apply l'Hopital's rule and find the limit as x approaches 1.
💡simplify
To 'simplify' in mathematics means to make a complex expression or equation more straightforward by reducing it to a less complex form while maintaining its equivalent value. In the video, simplification is an essential step in the process of evaluating the limit, as it involves reducing the complexity of the expression to a form that can be more easily analyzed or for which the limit can be more readily determined.
💡cancel out
To 'cancel out' in mathematics refers to the process of eliminating terms or factors that are equal but have opposite signs, resulting in a net value of zero. This process is often used in algebraic simplification and is crucial in the video for reducing the complexity of the expressions involved in finding the limit. In the context of the video, terms cancel out when simplifying the numerator and denominator after applying l'Hopital's rule, leading to a simpler form that can be used to evaluate the limit.
💡evaluation
In mathematics, 'evaluation' refers to the process of determining the value of an expression by substituting values for the variables and performing the necessary operations. In the video, evaluation is the goal of the process, where the aim is to find the value of the given expression as x approaches 1. This involves a series of steps, including algebraic manipulation and the application of calculus rules like l'Hopital's rule, to arrive at a determinate value for the limit.
Highlights

The problem involves finding the limit of a function as x approaches 1.

The initial attempt to evaluate the expression at x=1 results in an undefined form, 1/0 - 1/0.

The undefined form does not fit the criteria for l'Hopital's rule, which requires 0/0 or ∞/∞.

The speaker suggests not giving up and looking for a way to manipulate the expression to apply l'Hopital's rule.

By adding the two expressions, a common denominator is formed, which could potentially lead to the l'Hopital indeterminate form.

The algebraic manipulation results in a new expression that is equivalent to the original, but with a different form.

Upon evaluating the new expression at x=1, an indeterminate form of 0/0 is obtained, which is suitable for l'Hopital's rule.

The first derivative of the numerator is calculated using the product rule.

The derivative of the denominator is found by differentiating each term separately.

After simplifying, the expression can be rewritten in a form that is ready for the second application of l'Hopital's rule.

The second application of l'Hopital's rule leads to a simpler expression with a clear limit.

The final result of the limit, after applying l'Hopital's rule twice, is 1/2.

The process demonstrates the utility of algebraic manipulation in solving limits that do not initially present as indeterminate forms suitable for l'Hopital's rule.

The method showcases the importance of recognizing when to apply l'Hopital's rule and how to transform expressions to meet the criteria.

The example serves as a lesson in problem-solving, emphasizing persistence and creativity in mathematical analysis.

The transcript provides a step-by-step guide that can be followed for similar limit problems.

The approach taken can be generalized to a variety of limit problems, making it a valuable strategy in calculus.

Transcripts

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