Limits of trigonometric functions | Limits and continuity | AP Calculus AB | Khan Academy

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17 Jul 201706:05
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TLDRThe video script discusses the concept of limits involving trigonometric functions, emphasizing the continuity and definition across all real numbers for sine and cosine. It explains that the limit of these functions as x approaches a certain value is equal to the function's value at that point. However, for functions like tangent and cotangent, limits may not exist when approaching values that are not within their domain, such as x approaching Ο€/2, where the function is undefined, resulting in a vertical asymptote on the graph.

Takeaways
  • πŸ“Œ Trigonometric functions, specifically sine and cosine, are defined for all real numbers and are continuous over their entire domain.
  • 🌟 The limit of sine of x as x approaches Ο€ is equal to the value of sine at Ο€, which is 0.
  • πŸ“ˆ The limit of cosine of x as x approaches Ο€/4 is equal to the value of cosine at Ο€/4, which is √2/2 or approximately 0.7071.
  • πŸ”’ For limits involving sine or cosine, the limit as x approaches any real number a is equal to the sine or cosine of a.
  • πŸ₯‚ When considering limits of functions with constrained domains, such as tangent, the limit as x approaches Ο€ is equal to the value of the function at Ο€, which is 0/-1 or 0.
  • 🚫 The limit as x approaches Ο€/2 for the tangent function does not exist because Ο€/2 is not within the domain of the tangent function, resulting in an undefined expression.
  • πŸ“Š The domain of a function plays a crucial role in determining the existence of a limit. If the point is not in the domain, the limit may not exist.
  • πŸ”„ The cotangent function, being the reciprocal of the tangent function, also has a vertical asymptote at Ο€/2 and thus does not have a limit as x approaches Ο€/2.
  • 🧩 Understanding the behavior of trigonometric functions at their asymptotes is essential for evaluating limits and determining the function's continuity.
  • πŸ“ˆ The behavior of trigonometric functions at specific angles, such as Ο€/4 for cosine, is important for solving limit problems involving these functions.
Q & A

  • What is the domain of sine and cosine functions?

    -The domain of both sine and cosine functions is all real numbers, meaning they are defined for every real number value of x.

  • Is the sine function continuous over its domain?

    -Yes, the sine function is continuous over its entire domain, which includes all real numbers.

  • What is the value of the limit as x approaches Ο€ for the sine function?

    -The limit as x approaches Ο€ for the sine function is equal to the sine of Ο€, which is 0.

  • What is the value of the limit as x approaches Ο€/4 for the cosine function?

    -The limit as x approaches Ο€/4 for the cosine function is equal to the cosine of Ο€/4, which is √2/2 or approximately 0.7071.

  • What is the general rule for the limit of sine and cosine functions as x approaches a?

    -The general rule for the limit of sine and cosine functions as x approaches a is that the limit is equal to the sine or cosine of a, respectively.

  • What happens when the limit of a trigonometric function is taken at a point not in its domain?

    -If the limit of a trigonometric function is taken at a point not in its domain, the limit may not exist or be defined.

  • What is the domain of the tangent function?

    -The domain of the tangent function includes all real numbers except for the values where the cosine function is zero, which would create an undefined expression due to division by zero.

  • What is the limit as x approaches Ο€/2 for the tangent function?

    -The limit as x approaches Ο€/2 for the tangent function does not exist because Ο€/2 is not in the domain of the tangent function, as it would result in an undefined expression (infinity or undefined due to division by zero).

  • How can you identify if a limit of a trigonometric function exists?

    -A limit of a trigonometric function exists if the point at which the limit is taken is within the function's domain. If the point is not in the domain, the limit may not exist.

  • What is the cotangent function?

    -The cotangent function is the reciprocal of the tangent function, defined as the cosine of x divided by the sine of x.

  • What is the limit as x approaches Ο€ for the cotangent function?

    -The limit as x approaches Ο€ for the cotangent function is equal to the cotangent of Ο€, which is negative one divided by zero, resulting in an undefined value. However, this is represented as approaching negative infinity.

Outlines
00:00
πŸ“š Understanding Limits of Trigonometric Functions

This paragraph introduces the concept of limits in the context of trigonometric functions, specifically focusing on sine and cosine functions. It emphasizes that sine and cosine are defined for all real numbers and are continuous over their entire domain. The instructor illustrates the process of finding the limit as x approaches specific values, such as pi, by substituting these values into the functions. The key takeaway is that for any real number 'a', the limit of sine or cosine as x approaches 'a' is equal to the value of the function at 'a'. The paragraph also discusses the behavior of these functions when approaching undefined points, like pi/2 for the tangent function, which results in a vertical asymptote and no limit.

05:01
πŸ”’ Limits of Trigonometric Functions with Restricted Domains

The second paragraph delves into the behavior of trigonometric functions with restricted domains, such as tangent and cotangent. It explains that while the domain of sine and cosine includes all real numbers, functions like tangent and cotangent have certain limitations. The instructor uses the example of the limit as x approaches pi for the tangent function and demonstrates that it results in an undefined value due to division by zero. Similarly, the cotangent function has a vertical asymptote when x approaches pi, indicating that the limit does not exist. The key point is that if the value we're taking the limit to is within the domain of the function, the limit will be defined and equal to the function's value at that point. However, if it's not within the domain, as in the case of pi/2 for tangent or pi for cotangent, the limit does not exist.

Mindmap
Keywords
πŸ’‘limits
In the context of the video, limits refer to the value that a function approaches as the input (x) gets arbitrarily close to a certain value (a). It is a fundamental concept in calculus and is used to understand the behavior of functions, especially near points where the function may not be defined or where there is a sharp change in behavior. For example, the video discusses finding the limit as x approaches pi of the sine of x, which is equal to the sine of pi, demonstrating that the limit exists and is zero.
πŸ’‘trigonometric functions
Trigonometric functions are mathematical functions that relate angles to real numbers. The most common trigonometric functions are sine, cosine, and tangent. These functions are widely used in mathematics, physics, and engineering to model periodic phenomena. In the video, the focus is on the limits of trigonometric functions, especially how they behave as their input approaches certain values. The continuity and domain of these functions are crucial in determining whether the limits exist.
πŸ’‘sine function
The sine function, often abbreviated as sin, is one of the fundamental trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. In the video, the sine function is discussed in relation to its limit as x approaches pi, which is found to be zero. The sine function is also noted for being continuous and defined for all real numbers, which means it maintains a consistent value without breaks or jumps across its entire domain.
πŸ’‘cosine function
The cosine function, denoted as cos, is another primary trigonometric function. It is similar to the sine function but relates the angle to the ratio of the adjacent side's length to the hypotenuse's length in a right triangle. In the video, the cosine function's limit as x approaches pi/4 is discussed, which is equal to the square root of two divided by two. The cosine function, like the sine function, is continuous and defined for all real numbers.
πŸ’‘tangent function
The tangent function, written as tan, is a trigonometric function that equals the ratio of the sine function to the cosine function, or sin(x)/cos(x). It is used to find the angle of a right triangle when the ratio of the opposite side to the adjacent side is known. In the video, the limit of the tangent function as x approaches pi is explored, which turns out to be zero, as the sine of pi is zero and the cosine of pi is negative one, resulting in a zero-over-negative-one scenario.
πŸ’‘domain
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In the context of the video, understanding the domain of trigonometric functions is crucial for determining the existence of limits. For instance, while sine and cosine have a domain of all real numbers, the tangent function has a domain that excludes values where the cosine function is zero, as these would lead to division by zero, making the function undefined.
πŸ’‘continuous
A function is said to be continuous at a point if it exists at that point and its graph can be drawn without lifting the pen from the paper. In the video, the continuity of trigonometric functions like sine and cosine is highlighted, as they are defined for all real numbers and have no breaks or jumps in their graphs. This property is essential for the existence of limits because a function can only have a limit at a point if it is continuous there.
πŸ’‘asymptote
An asymptote is a line that a graph approaches but never actually intersects, even though the distance between the graph and the line becomes arbitrarily small. In the context of the video, vertical asymptotes are mentioned for functions like the tangent and cotangent when x approaches pi/2. These asymptotes indicate points at which the function is not defined, and thus, the limits do not exist at those points.
πŸ’‘cosecant function
The cosecant function, denoted as csc, is the reciprocal of the sine function, or 1/sin(x). It is one of the six basic trigonometric functions and is used to find the angle when the sine value is known. In the video, the cosecant function is mentioned in the context of limits, specifically when x approaches pi/2, where the function is not defined because the sine function, which it depends on, is zero, leading to an undefined value.
πŸ’‘cotangent function
The cotangent function, often abbreviated as cot or tan^-1, is the reciprocal of the tangent function, or cos(x)/sin(x). It is used to find the angle when the ratio of the adjacent side to the opposite side in a right triangle is given. In the video, the cotangent function's limit as x approaches pi is discussed, which does not exist because pi is not within the domain of the cotangent function, as substituting pi would result in division by zero.
πŸ’‘limit points
Limit points are values of x for which the limit of a function exists. In the video, the concept is used to discuss the behavior of trigonometric functions as x approaches certain values. It is emphasized that if the limit point is within the domain of the function, the limit will exist and be equal to the function's value at that point. However, if the limit point is not in the domain, as with pi/2 for the tangent function, the limit does not exist.
Highlights

The video discusses limits involving trigonometric functions, focusing on understanding their behavior as they approach certain values.

Sine and cosine functions are defined for all real numbers and are continuous over their entire domain.

The limit of sine of x as x approaches pi is equal to the value of sine at pi, which is zero.

For cosine of x, the limit as x approaches pi over four is equal to the square root of two over two, corresponding to a 45-degree angle.

The general rule for sine and cosine is that the limit as x approaches a is equal to the sine or cosine of a, provided a is within their domain.

The tangent function, being a ratio of sine to cosine, is not defined for all real numbers due to potential division by zero.

The limit of tangent of x as x approaches pi does exist and is equal to zero, as the denominator (cosine of pi) is negative one, not zero.

The limit of tangent of x as x approaches pi over two does not exist, as cosine of pi over two is zero, leading to an undefined expression.

For cotangent of x, which is the reciprocal of tangent of x, the limit as x approaches pi does not exist because pi is not in its domain.

The concept of vertical asymptotes in trigonometric functions is introduced, where the function approaches but never reaches a vertical line.

The video emphasizes the importance of the domain of a function when considering limits, as limits to points not in the domain may not exist.

The video provides a clear and structured approach to understanding and calculating limits of trigonometric functions.

Trigonometric functions' continuity and their behavior at undefined points are crucial for determining the existence of limits.

The video serves as a valuable educational resource for learners seeking to understand the properties and applications of limits in trigonometry.

The practical application of the concepts discussed in the video can be seen in various fields such as physics, engineering, and mathematics.

The video's content is a testament to the mathematical rigor and precision required when analyzing the behavior of functions and their limits.

Transcripts

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