Trig limit using pythagorean identity | Limits and continuity | AP Calculus AB | Khan Academy

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13 Jul 201607:01

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TLDRThe video script explores the mathematical concept of finding the limit as theta approaches zero for the expression (1 - cos(theta)) / (2 * sin^2(theta)). Initially, direct substitution results in an indeterminate form of zero over zero. The video then introduces a trigonometric identity to rewrite the expression and resolve the indeterminate form. Through algebraic manipulation and factoring, the expression is transformed into a form that allows for direct substitution at theta equals zero, ultimately yielding a limit of one fourth.

Takeaways

📌 The problem involves finding the limit of a trigonometric expression as theta approaches zero.
🔍 Initially, one might be tempted to directly substitute theta with zero, but this leads to an indeterminate form (zero over zero).
🧩 To resolve the indeterminate form, the script suggests looking for alternative approaches rather than giving up on finding the limit.
📐 The concept of continuity is highlighted; if a function is continuous at a point, its limit at that point equals the function's value at that point.
🌀 The script introduces the use of trigonometric identities, specifically the Pythagorean identity, to simplify the given expression.
🔄 By rewriting the expression using the Pythagorean identity, the script transforms the original indeterminate form into a more manageable one.
📂 The technique of factoring the expression as a difference of squares is used to further simplify the calculation.
🔄 The script emphasizes the importance of mathematical clarity when manipulating expressions and the need to maintain the same domain for the function.
🔧 A new function, G of X, is introduced that is continuous at zero and equivalent to the original function, F of X, except at the point of discontinuity.
🎯 The limit of G of X as theta approaches zero is found by direct substitution, since G of X is continuous at zero.
🏁 The final answer to the limit problem is calculated to be one fourth (1/4).

Q & A

What is the main topic of the video script?
-The main topic of the video script is finding the limit of a mathematical expression as theta approaches zero, specifically the limit of (1 - cos(theta)) / (2 * sin^2(theta)).
Why can't we directly substitute theta equals zero into the original expression?
-We can't directly substitute theta equals zero into the original expression because doing so results in an indeterminate form of zero over zero, which does not provide a clear value for the limit.
How does the video suggest we should approach the indeterminate form?
-The video suggests that when faced with an indeterminate form like zero over zero, we should not give up. Instead, we should look for alternative approaches or methods to simplify the expression so that the limit can be determined.
What trigonometric identity is used to rewrite the expression and avoid the indeterminate form?
-The Pythagorean identity from trigonometry, which states that sin^2(theta) + cos^2(theta) equals one, is used to rewrite the expression and avoid the indeterminate form.
How is the expression rewritten using the Pythagorean identity?
-The expression is rewritten as (1 - cos(theta)) / (2 * (1 - cos^2(theta))) by replacing sin^2(theta) with 1 - cos^2(theta) according to the Pythagorean identity.
What algebraic technique is used to further simplify the rewritten expression?
-The algebraic technique of factoring a difference of squares is used to further simplify the rewritten expression, resulting in (1 - cos(theta)) / ( (1 + cos(theta)) * (1 - cos(theta)) ).
Why is it important to be clear about the mathematical steps when finding the limit?
-It is important to be clear about the mathematical steps when finding the limit to ensure that the process is logically sound and that the resulting limit is valid. This clarity helps avoid mistakes and maintains the integrity of the mathematical function and its domain.
How does the video define a new function, G of X, and why?
-The video defines a new function, G of X, as one over two plus two times cosine theta, which is continuous at zero and identical to F of X except at theta equals zero. This new function is defined to bypass the point of discontinuity in F of X, allowing for a direct substitution to find the limit.
What is the limit of G of X as theta approaches zero?
-The limit of G of X as theta approaches zero is one fourth, which is found by substituting zero for theta in the expression for G of X, resulting in one over two plus two times cosine of zero, which simplifies to one fourth.
What is the key takeaway from the video script for finding limits in similar mathematical scenarios?
-The key takeaway from the video script is that when faced with an indeterminate form while finding limits, one should explore alternative methods to simplify the expression and find a continuous function with the same domain. This approach allows for a valid determination of the limit through direct substitution.

Outlines

00:00

📚 Introduction to the Limit Problem

The first paragraph introduces a mathematical problem involving limits. The problem is to find the limit of the expression (1 - cos(θ))/2sin^2(θ) as θ approaches zero. The voiceover suggests pausing the video for viewers to attempt the problem themselves. It then explains that one might be tempted to directly substitute θ with zero, but this leads to an indeterminate form of 0/0. The speaker emphasizes that having an indeterminate form doesn't mean the limit doesn't exist; it suggests looking for alternative approaches to solve the problem. The paragraph ends with the speaker considering using trigonometric identities to simplify the expression and avoid the indeterminate form when θ approaches zero.

05:00

🔍 Solving the Indeterminate Form and Finding the Limit

In the second paragraph, the speaker addresses the indeterminate form 0/0 obtained in the previous paragraph by utilizing the Pythagorean identity from trigonometry, which states that sin^2(θ) + cos^2(θ) = 1. By applying this identity, the expression is rewritten as (1 - cos(θ))/(2(1 - cos^2(θ))). The speaker then factors the denominator as a difference of squares, leading to the expression (1 - cos(θ))/((1 + cos(θ))(1 - cos(θ))). This allows the speaker to introduce a new function, G(θ), which is equivalent to F(θ) except it's defined at θ = 0. The speaker then finds the limit of G(θ) as θ approaches zero by substituting zero into the function, resulting in a well-defined limit of 1/4. The paragraph concludes with the speaker emphasizing the importance of mathematical clarity and ensuring that any simplifications or substitutions do not alter the domain of the function being considered.

Mindmap

Keywords

💡limit

In the context of the video, 'limit' refers to the value that a function or sequence approaches as the input (in this case, theta) gets arbitrarily close to a certain point (in this case, zero). It is a fundamental concept in calculus and mathematical analysis, used to describe the behavior of functions at the boundaries of their domains. The video discusses finding the limit of a specific mathematical expression as theta approaches zero.

💡cosine

The term 'cosine' is a trigonometric function that relates the angle of a right triangle to the ratio of the length of its adjacent side to its hypotenuse. In the video, the cosine function is used in the expression that the limit is being taken of, and the speaker discusses the behavior of cosine as theta approaches zero, where the cosine of zero is one.

💡sine

The 'sine' function, like cosine, is a trigonometric function that relates the angle of a right triangle to the ratio of the length of its opposite side to its hypotenuse. The video mentions sine in the context of the expression for which the limit is being found, and it highlights that the sine of zero is zero, leading to an indeterminate form when squared.

💡indeterminate form

An 'indeterminate form' occurs when simplifying expressions leads to an algebraic situation where the result is undefined, such as 0/0. In the video, the limit initially results in an indeterminate form, which prompts the speaker to seek alternative methods for finding the limit. This concept is crucial for solving limits in calculus when direct substitution leads to an undefined expression.

💡Pythagorean Identity

The 'Pythagorean Identity' is a fundamental relationship in trigonometry stating that for any angle, the square of the sine of the angle plus the square of the cosine of the angle is equal to one. In the video, this identity is used to rewrite the given expression in a form that helps resolve the indeterminate form and find the limit as theta approaches zero.

💡factoring

In mathematics, 'factoring' refers to the process of breaking down a polynomial into a product of other polynomials or factors. The video uses factoring to simplify the expression for the limit by viewing it as a difference of squares, which allows the speaker to rewrite the expression in a form that avoids the indeterminate form and leads to the solution.

💡continuous

A function is 'continuous' at a point if it is defined at that point and its graph does not have any breaks or jumps at that point. In the video, the concept of continuity is important for understanding the behavior of the function as theta approaches zero and for finding equivalent expressions that can be used to evaluate the limit without encountering discontinuities.

💡domain

The 'domain' of a function is the set of all possible input values (in this case, theta) for which the function is defined. The video discusses the need to ensure that the functions being compared have the same domain, except at the point of interest (zero), to validly compare their limits.

💡substitution

In mathematics, 'substitution' is the process of replacing one expression with another, often to simplify an equation or to evaluate a function at a specific point. The video emphasizes the importance of being clear about the domain of the function when substituting theta with zero to find the limit.

💡theta

In the context of the video, 'theta' represents the variable input to the trigonometric functions and the subject of the limit calculation. The video focuses on finding the limit of a function as theta approaches zero, which is a common point of interest in calculus for evaluating trigonometric limits.

💡algebra

The term 'algebra' refers to a branch of mathematics that uses symbols and rules to solve equations and analyze mathematical structures. In the video, algebraic manipulation is used to simplify the expression and find the limit, showcasing the importance of algebra in understanding and solving mathematical problems.

Highlights

The problem presented involves finding the limit of a trigonometric expression as theta approaches zero.

The initial approach to the problem is to directly evaluate the expression at theta equals zero, resulting in an indeterminate form of zero over zero.

Despite the indeterminate form, it is emphasized that the limit might still exist, and alternative approaches should be considered.

The use of trigonometric identities, specifically the Pythagorean identity, is suggested as a method to simplify the expression and avoid the indeterminate form.

The expression is rewritten using the Pythagorean identity, resulting in a new form that is a difference of squares.

The concept of factoring the expression as a difference of squares is introduced to further simplify the problem.

The factored expression reveals a common term in the numerator and denominator, which can be exploited for simplification.

A new function, G of X, is defined to be continuous at theta equals zero, unlike the original function F of X.

It is established that the limits of F of X and G of X as theta approaches zero will be the same, despite the difference in their definitions at zero.

The importance of mathematical clarity when manipulating expressions and functions is emphasized to maintain the same domain.

The limit of G of X is calculated by direct substitution of zero for theta, bypassing the need for complex algebraic manipulation.

The final answer to the limit problem is found to be one fourth, demonstrating a straightforward solution through the use of continuous functions.

The transcript provides a detailed walkthrough of a limit problem involving trigonometric functions, showcasing a methodical approach to problem-solving.

The explanation highlights the utility of trigonometric identities in simplifying complex expressions and finding limits.

The process emphasizes the importance of understanding the domain of functions when working with limits and continuity.

The solution path avoids potential pitfalls in handling indeterminate forms by seeking alternative expressions that are better behaved.

Transcripts

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Takeaways

Q & A

What is the main topic of the video script?

Why can't we directly substitute theta equals zero into the original expression?

How does the video suggest we should approach the indeterminate form?

What trigonometric identity is used to rewrite the expression and avoid the indeterminate form?

How is the expression rewritten using the Pythagorean identity?

What algebraic technique is used to further simplify the rewritten expression?

Why is it important to be clear about the mathematical steps when finding the limit?

How does the video define a new function, G of X, and why?

What is the limit of G of X as theta approaches zero?

What is the key takeaway from the video script for finding limits in similar mathematical scenarios?