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

Khan Academy
13 Jul 201610:30
EducationalLearning
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TLDRThis video script delves into the process of finding the limit of a trigonometric expression as theta approaches negative pi over 4. The key lies in recognizing and handling the indeterminate form of 0/0, using algebraic manipulation and trigonometric identities to simplify the expression. The video emphasizes the importance of precise mathematical reasoning and the concept that if two functions are equal for all values except a point where the original function is undefined but the new function is continuous, their limits at that point will be the same. The final answer obtained for the limit is 1/2.

Takeaways
  • πŸ” The task is to find the limit of a trigonometric expression as theta approaches negative pi over 4.
  • πŸ“Œ Initially, the expression 1/√(2sinΞΈ) over cos(2ΞΈ) is considered, with an attempt to evaluate it directly.
  • 🧠 A substitution approach is suggested, rewriting the expression to make the limit calculation easier.
  • πŸ”’ The sine and cosine values at theta equals negative pi over 4 are used to simplify the expression.
  • πŸŒ€ An indeterminate form of 0/0 is encountered, indicating the need for further algebraic manipulation.
  • πŸ“ Trigonometric identities, particularly those involving cos(2ΞΈ), are introduced to help resolve the indeterminate form.
  • πŸ”„ The expression is factored to cancel out terms, leading to a new equivalent expression to evaluate the limit.
  • 🚫 It's emphasized that the new expression must have the same domain restrictions as the original for the limit to be valid.
  • πŸ“ˆ The limit is re-evaluated with the new expression, yielding a result of 1/2.
  • πŸ‘‰ The distinction between the original and the new expression is clarified, highlighting the importance of domain and continuity.
  • 🎯 The key takeaway is that if two expressions are equal for all values except where one is not defined, and the other is continuous, their limits at that point are equal.
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 negative pi over 4.

  • What is the expression given in the script?

    -The expression given in the script is 1 over the square root of 2 sine of theta, divided by cosine of 2 theta.

  • What is the initial approach taken to find the limit?

    -The initial approach taken to find the limit is to substitute theta with negative pi over 4 directly into the expression and see if the limit can be calculated.

  • Why can't we directly substitute theta with negative pi over 4?

    -We can't directly substitute theta with negative pi over 4 because it results in an indeterminate form of zero over zero, which is undefined.

  • What is the significance of the trig identities in solving this problem?

    -The trig identities are significant in solving this problem because they allow us to algebraically manipulate the expression to cancel out terms and resolve the indeterminate form.

  • How is the expression transformed to avoid the indeterminate form?

    -The expression is transformed by using the identity for cosine of 2 theta, which is rewritten as a difference of squares, and then factored to cancel out terms with the square root of 2 sine theta.

  • What is the domain restriction mentioned in the script?

    -The domain restriction mentioned is that theta should not be equal to negative pi over 4 for the transformed expression to be equivalent to the original one.

  • What is the final answer for the limit?

    -The final answer for the limit is 1/2.

  • Why is it important to recognize that the original expression and the transformed expression are not the same?

    -It is important to recognize that the original expression and the transformed expression are not the same because the transformed expression is only equal to the original for all values of theta except where the original was undefined. This understanding is crucial for correctly applying limit concepts.

  • What is the key takeaway from the script regarding limits?

    -The key takeaway is that if two expressions are equal for all values of x except at a point where the original was not defined, and the new expression is continuous at that point, then their limits as x approaches that point are equal.

Outlines
00:00
πŸ“š Understanding Limits and Trig Identities

The paragraph begins with an exploration of the limit of a mathematical function as theta approaches negative pi over 4. The voiceover suggests the viewer attempt the problem before proceeding. The approach involves comparing the limit of the original function to a transformed version, using the properties of continuous functions. The explanation delves into the specifics of the trigonometric functions at theta equals negative pi over 4, highlighting the importance of recognizing the difference between the original and transformed expressions. The discussion also touches on the use of trigonometric identities, particularly those involving cosine of 2 theta, to simplify and solve the limit problem. The paragraph emphasizes the need for algebraic manipulation to resolve the indeterminate form of zero over zero, ultimately leading to the cancellation of terms and a defined limit.

05:02
πŸ” Factoring and Simplification for Limits

This paragraph focuses on the process of factoring and simplifying the given trigonometric expression to find the limit as theta approaches negative pi over 4. The voiceover explains the cancellation of terms and how it leads to a simpler expression. The discussion also addresses the importance of considering the domain of the function, noting that the original expression is not defined at theta equals negative pi over 4, while the transformed expression is. The summary underlines the significance of being precise and understanding the difference between the two expressions, even though they are equal for all values of theta within a specified interval. The paragraph concludes with a reiteration of the importance of continuity and the equivalence of limits for functions that are equal for all x except at a certain point where one is continuous.

10:06
🎯 Final Answer and Recap of Limit Concepts

The final paragraph wraps up the discussion on the limit of the trigonometric expression and provides the answer, which is 1/2. It reiterates the main points from the previous paragraphs, emphasizing the process of algebraic manipulation and the use of trigonometric identities to simplify the expression and find the limit. The paragraph also reinforces the concept that if two functions are equal for all x except at a certain point where one is continuous, their limits at that point will be equal. The summary serves as a recap of the key concepts and methods used to solve the limit problem, ensuring the viewer has a clear understanding of the process and the final result.

Mindmap
Keywords
πŸ’‘limit
In the context of the video, 'limit' refers to a fundamental concept in calculus that describes the value that a function or sequence 'approaches' as the input (theta in this case) gets arbitrarily close to a certain point. The video specifically focuses on finding the limit as theta approaches negative pi over 4 for a given trigonometric expression. The concept is crucial for understanding the behavior of functions near specific points.
πŸ’‘square root
A 'square root' is a mathematical operation that finds the non-negative number which, when multiplied by itself, gives the original number. In the video, the square root is applied to the number 2 and is part of the trigonometric expression being analyzed. The square root is a key component in the algebraic manipulations and the final solution.
πŸ’‘sine
The 'sine' function is one of the six trigonometric functions, which relates the angles of a right triangle to the ratios of its sides. In the video, the sine function is used in the context of the expression being evaluated and is a core element in the trigonometric calculations, especially when dealing with the value of sine at negative pi over 4.
πŸ’‘cosine
The 'cosine' function, like 'sine', is a trigonometric function that relates the angles of a right triangle to the ratios of its sides. It is used in the video to express the denominator of the given expression and is crucial for the limit calculation. The video also discusses the cosine of 2 theta and its relationship with other trigonometric identities.
πŸ’‘theta
In the context of the video, 'theta' represents the variable in the trigonometric expressions and functions being analyzed. Theta is a common symbol used in mathematics to denote angles, and its value is crucial in determining the behavior of functions and the evaluation of limits.
πŸ’‘trigonometric identities
Trigonometric identities are equations that are true for all values of the angles they contain. They are fundamental in trigonometry and are used to simplify expressions and solve problems involving trigonometric functions. In the video, identities like 'cosine of 2 theta' are used to manipulate and simplify the given expression to find the limit.
πŸ’‘indeterminate form
An 'indeterminate form' arises in limit calculations when the expression simplifies to a ratio of two terms that are both zero, or undefined, such as 0/0. This does not necessarily mean that the limit does not exist; it often indicates that further algebraic manipulation or a different approach is needed to evaluate the limit. The video addresses this concept when the initial expression simplifies to 'zero over zero'.
πŸ’‘algebraic manipulation
Algebraic manipulation refers to the process of transforming and rearranging mathematical expressions using algebraic rules and operations. In the context of the video, algebraic manipulation is used to change the original expression into a form that avoids the indeterminate form and allows for the evaluation of the limit.
πŸ’‘continuous function
A 'continuous function' is one where there are no breaks, jumps, or holes in its graph. In the context of the video, the continuity of a function is important when considering the substitution of values and the behavior of the function near certain points. The video ensures that the function being analyzed is continuous at the point of interest, which is crucial for the limit calculation.
πŸ’‘open interval
An 'open interval' in mathematics is a set of real numbers that does not include its endpoints. In the video, the concept of an open interval is used to define the domain of the function where the function is continuous and defined, except for certain points where it may not be defined or continuous.
πŸ’‘Pythagorean identity
The 'Pythagorean identity' is a fundamental trigonometric identity derived from the Pythagorean theorem. It relates the sine and cosine functions of the same angle through the equation 'sin^2(theta) + cos^2(theta) = 1'. In the video, this identity is used to transform and simplify expressions involving sine and cosine.
Highlights

The task is to find the limit of a mathematical expression as theta approaches negative pi over 4.

The expression involves the square root of 2 sine of theta over cosine of 2 theta.

The initial approach is to compare the given expression with a similar one for the purpose of substitution.

The trigonometric values are assumed to be in radians, with negative pi over 4 corresponding to a negative 45-degree angle.

The sine of negative pi over 4 is negative square root of 2 over 2.

The limit initially results in an indeterminate form of zero over zero.

Indeterminate forms can be resolved through algebraic manipulation using trigonometric identities.

Cosine of 2 theta can be expressed in multiple ways using trigonometric identities, which might help in simplifying the expression.

The goal is to cancel out terms that lead to the indeterminate form of zero over zero.

The expression can be factored to reveal a potential for cancellation, leading to a simplified form.

The simplified form of the expression avoids the undefined point at theta equals negative pi over 4.

The continuity of the new expression at theta equals negative pi over 4 is crucial for determining the limit.

The limit of the new expression is found to be 1/2 by algebraic simplification.

It's important to recognize that the original and the simplified expressions are not the same, except in the open interval around negative pi over 4.

The concept of limits and continuity is emphasized in the process of solving the problem.

The final answer to the limit of the given expression as theta approaches negative pi over 4 is 1/2.

Transcripts

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TrigonometryLimitsAlgebraic ManipulationTrig IdentitiesTheta FunctionsContinuous FunctionsIndeterminate FormsSine and CosineMathematical AnalysisEducational Content