Limits of Trigonometric Functions

The Organic Chemistry Tutor
20 Feb 201815:23
EducationalLearning
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TLDRThis transcript discusses various limit problems involving trigonometric functions as x approaches 0. It explains how to confirm the limit of sin(x)/x as 1 by substituting values close to zero and demonstrates the use of substitution and known limit formulas to solve problems like the limit of sin(5x)/x and sin(2x)/5x. The process is also shown for more complex expressions, such as sin(7x)/sin(3x) and tan(x)/x, emphasizing the importance of showing work for a clear understanding of the concepts.

Takeaways
  • πŸ“ˆ The limit of sin(x)/x as x approaches 0 is 1, which can be confirmed by plugging in values close to 0 into a calculator.
  • πŸ”’ For the limit of (1 - cos(x))/x as x approaches 0, the value is 0, demonstrated by plugging in small values like 0.1 and 0.01.
  • 🌟 When calculating limits involving sin(nx)/x as x approaches 0, the answer is n, as shown by the example of sin(5x)/x yielding 5.
  • πŸ“š To find the limit of sin(2x)/5x as x approaches 0, the process involves multiplying the numerator and denominator by 2/5 and simplification yields 2/5.
  • πŸ”„ For limits involving sin(7x)/sin(3x) as x approaches 0, the result is 7/3 after separating and simplifying the expression using substitution.
  • 🧭 The limit of tan(x)/x as x approaches 0 is 1, which can be deduced by rewriting tan(x) as sin(x)/cos(x) and applying known limits.
  • πŸ”’ For the limit of tan(4x)/3x as x approaches 0, the answer is 4/3, following a similar process of substitution and simplification.
  • βœ… The limit of (7 * (1 - cos(x)))/x as x approaches 0 is 0, since the limit of (1 - cos(x))/x is 0.
  • πŸ“ The limit of sin^2(x)/x as x approaches 0 is 0, as sin(x)/x approaches 1 and x approaches 0, resulting in a product of 0.
  • πŸŒ€ For the limit of sin(x^2)/x as x approaches 0, the process involves multiplying the numerator and denominator by x and using substitution with y = x^2 to find the limit is 0.
Q & A
  • What is the value of the limit as x approaches 0 of sine x divided by x?

    -The value of the limit as x approaches 0 of sine x divided by x is 1.

  • How can you confirm the limit using a calculator?

    -You can confirm the limit by plugging in a number very close to 0 into the calculator, ensuring it's in radian mode, and then dividing the sine of that number by the number itself. As the input approaches zero, the result should approach the limit's value.

  • What is the formula for the limit as x approaches 0 of (1 - cosine x) divided by x?

    -The formula for the limit as x approaches 0 of (1 - cosine x) divided by x is 0.

  • How does the value of the limit as x approaches 0 of (1 - cosine x) divided by x demonstrate its truth?

    -By plugging in values close to 0, such as 0.1 and 0.01, and observing that the results approach 0, we can demonstrate that the limit as x approaches 0 of (1 - cosine x) divided by x is indeed 0.

  • What is the limit as x approaches 0 of sine 5x divided by x?

    -The limit as x approaches 0 of sine 5x divided by x is 5. This is derived by multiplying the numerator and denominator by 5 and then using the known limit of sine x over x as x approaches 0, which is 1.

  • How can you evaluate the limit as x approaches 0 of sine 2x divided by 5x?

    -The limit as x approaches 0 of sine 2x divided by 5x is 2 divided by 5 or 0.4. This is shown by multiplying the numerator and denominator by 2/2 and then using the known limit of sine x over x as x approaches 0.

  • What is the limit as x approaches 0 for sine 7x divided by sine 3x?

    -The limit as x approaches 0 for sine 7x divided by sine 3x is 7 divided by 3. This is because when simplifying the expression and using the known limit of sine x over x as x approaches 0, the ratio of the coefficients (7 and 3) remains.

  • What is the value of the limit as x approaches 0 of tangent x divided by x?

    -The value of the limit as x approaches 0 of tangent x divided by x is 1. Since tangent x is sine x divided by cosine x, and the limit of sine x over x as x approaches 0 is 1, and the limit of cosine x as x approaches 0 is 1, the result is 1.

  • How do you find the limit as x approaches 0 of tan 4x divided by 3x?

    -The limit as x approaches 0 of tan 4x divided by 3x is 4 divided by 3. This is found by converting tangent to sine over cosine, multiplying the numerator and denominator by 4/4, and then using the known limit of sine x over x as x approaches 0.

  • What is the result of the limit as x approaches 0 of seven times one minus cosine x divided by x?

    -The result of the limit as x approaches 0 of seven times one minus cosine x divided by x is 0. This is because the limit of one minus cosine x divided by x as x approaches 0 is 0, and any number times zero is zero.

  • How does the limit as x approaches 0 of sine squared x divided by x evaluate?

    -The limit as x approaches 0 of sine squared x divided by x evaluates to 0. This is because sine squared x is sine x times sine x, and the limit of sine x over x as x approaches 0 is 1 while the limit of x as x approaches 0 is 0, resulting in 1 times 0, which is 0.

  • What is the limit as x approaches 0 of sine of x squared divided by x?

    -The limit as x approaches 0 of sine of x squared divided by x is 0. This is found by multiplying the numerator and denominator by x to get the limit of sine x squared over x squared, and then using the limits as x approaches 0, which are 1 for sine x squared over x squared and 0 for x, resulting in 0.

Outlines
00:00
πŸ“š Introduction to Trigonometric Limits

This paragraph introduces the concept of limits with trigonometric functions, focusing on the limit of sine x divided by x as x approaches 0, which is equal to 1. The explanation includes a practical technique using a calculator to approximate the limit by plugging in values close to zero. It also introduces another formula, the limit of (1 - cosine x) divided by x as x approaches 0, which equals 0, and demonstrates its confirmation through similar approximation methods.

05:02
πŸ”’ Solving Trigonometric Limit Problems

The paragraph delves into solving specific trigonometric limit problems, such as the limit of sine 5x divided by x as x approaches 0, which simplifies to 5. It explains the process of substitution and demonstrates how to manipulate the expression to find the limit. The paragraph also covers the limit of sine 2x divided by 5x as x approaches 0, showing the steps to arrive at the answer of 2/5. Additionally, it touches on the limit of sine 7x divided by sine 3x as x approaches 0, which simplifies to 7/3, with a detailed explanation of the substitution and rearrangement of terms.

10:05
πŸ“ˆ Understanding Tangent and Sine Limits

This section discusses the limit of tangent x divided by x as x approaches 0, revealing that the answer is 1. It explains the relationship between tangent and sine/cosine, and how to apply the limit of sine x over x to find the solution. The paragraph also explores the limit of tan 4x divided by 3x as x approaches 0, confirming the answer as 4/3 through a step-by-step demonstration. Furthermore, it addresses the limit of (7 * (1 - cosine x)) divided by x as x approaches 0, which simplifies to 0, and the limit of sine squared x divided by x, which results in 0.

15:08
🧩 Advanced Trigonometric Limit Examples

The final paragraph presents more complex examples of trigonometric limits. It starts with the limit of sine of x squared divided by x as x approaches 0, which requires multiplying the numerator and denominator by x and using substitution to find the limit. The explanation then moves on to the limit of sine x squared divided by x, where the process involves separating the expression into two limits and simplifying to reach the answer of 0. The paragraph concludes with the limit of sine y divided by y times the limit of x as x approaches 0, which simplifies to 0, demonstrating the application of substitution and the properties of limits.

Mindmap
Keywords
πŸ’‘Limits
Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a particular value. In the context of the video, limits are used to evaluate what happens to trigonometric functions as 'x' approaches zero. For example, the limit of the sine function divided by 'x' as 'x' approaches zero is equal to 1, which is a basic limit used to solve more complex problems in the video.
πŸ’‘Trigonometric Functions
Trigonometric functions, including sine, cosine, and tangent, are mathematical functions that relate angles to real numbers. These functions are essential in solving problems involving periodic phenomena. In the video, trigonometric functions are used in various limit calculations, such as sin(x)/x and tan(x)/x, to determine their behavior as 'x' approaches zero.
πŸ’‘Substitution
Substitution is a mathematical technique used to simplify expressions or to solve equations by replacing one or more variables with equivalent expressions. In the video, substitution is used as a method to simplify limits by replacing 'x' with a new variable, such as 'y', which allows for easier evaluation of the limit as 'x' approaches zero.
πŸ’‘Direct Substitution
Direct substitution is a method used in calculus to find the value of a function at a particular point by simply plugging in the value of the independent variable. In the context of the video, direct substitution is used to evaluate limits where the limit can be determined by simply replacing 'x' with the value it approaches.
πŸ’‘Sine Function
The sine function is a trigonometric function that represents the ratio of the length of the side opposite to an angle in a right triangle to the length of the hypotenuse. It is periodic with a period of 2Ο€ and oscillates between -1 and 1. In the video, the sine function is used extensively in calculating limits, such as the limit of sin(x)/x as x approaches 0, which is a fundamental limit equal to 1.
πŸ’‘Cosine Function
The cosine function is a trigonometric function that represents the ratio of the length of the adjacent side to an angle in a right triangle to the length of the hypotenuse. Like the sine function, it is also periodic with a period of 2Ο€ and oscillates between -1 and 1. In the video, the cosine function is used in limit calculations, such as the limit of (1 - cos(x))/x as x approaches 0, which equals 0.
πŸ’‘Tangent Function
The tangent function is a trigonometric function that represents the ratio of the sine function to the cosine function, or sin(x)/cos(x), at a given angle. It is used to find the slope of a line at a point on a curve and is periodic with a period of Ο€. In the video, the tangent function is discussed in the context of limits, such as the limit of tan(x)/x as x approaches 0, which equals 1.
πŸ’‘Calculator
A calculator is an electronic device used to perform mathematical calculations. In the context of the video, a calculator is suggested as a tool to approximate limits by plugging in values close to the point at which the limit is being evaluated, particularly when dealing with limits that involve trigonometric functions.
πŸ’‘Radian Mode
Radian mode is a setting on scientific calculators that allows for the correct evaluation of trigonometric functions in radians rather than degrees. It is important to use the correct mode when calculating limits involving trigonometric functions, as the values can differ significantly between radians and degrees, especially as the input approaches zero.
πŸ’‘Factoring
Factoring is the process of breaking down a polynomial into its constituent factors, which are simpler expressions that when multiplied together yield the original polynomial. In the context of the video, factoring is not explicitly mentioned but is a mathematical technique that could be used in simplifying expressions before evaluating limits.
πŸ’‘Direct Calculation
Direct calculation refers to the process of computing a value by applying the rules of arithmetic or algebra without the need for approximation or substitution. In the video, direct calculation is used when evaluating limits that can be simplified to a straightforward numerical value, such as the limit of x as x approaches 0.
Highlights

The limit of sine x divided by x as x approaches 0 is equal to 1.

Using a calculator in radian mode can confirm the limit by plugging in values close to 0, such as 0.1 and 0.01.

The limit of 1 minus cosine x divided by x as x approaches 0 is equal to 0.

Substitution technique involves multiplying the numerator and denominator by the same value to simplify the limit expression.

The limit of sine 5x divided by x as x approaches 0 can be found by multiplying the expression by 5 and using substitution.

The limit of sine 2x divided by 5x as x approaches 0 simplifies to 2 divided by 5 by using substitution and the known limit of sine x over x.

The limit of sine 7x divided by sine 3x as x approaches 0 is seven divided by three by separating the terms and using substitution.

The limit of tangent x divided by x as x approaches 0 is 1, as tangent is sine over cosine and the limit of sine x over x is 1.

For the limit of tan 4x divided by 3x as x approaches 0, the answer is 4 over 3, which is derived by converting tangent to sine over cosine and applying limits.

The limit of 7 times one minus cosine x divided by x as x approaches 0 is zero, as the limit of one minus cosine x divided by x is 0.

The limit of sine squared x divided by x as x approaches 0 is 0, by separating it into two limits and multiplying by 0.

The limit of sine of x squared divided by x as x approaches 0 is zero, using substitution and the fact that the limit of x as x approaches 0 is 0.

Trigonometric limits can be evaluated using direct substitution, calculator approximation, and substitution techniques.

Understanding the behavior of trigonometric functions near zero is crucial for solving limit problems.

The concept of limits allows us to determine the value a function approaches as the input gets arbitrarily close to a certain point.

The use of trigonometric identities and properties can simplify the process of finding limits.

The limit of a function can be found by breaking down complex expressions into simpler components.

The technique of substitution is particularly useful when dealing with trigonometric functions involving multiples of the variable.

The concept of limits is fundamental in calculus and is used to understand the behavior of functions at specific points.

Transcripts
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