Analyzing functions for discontinuities (discontinuity example) | AP Calculus AB | Khan Academy

Khan Academy
15 Jul 201604:02
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TLDRThe video script discusses a piecewise continuous function, defined differently for x values less than or equal to 2 and greater than 2. The focus is on finding the limit of the function as x approaches 2. It explains that the limit requires evaluating the left-hand and right-hand limits, which are found to be the natural log of 2 and four times the natural log of 2, respectively. However, the script reveals a discontinuity due to these limits being unequal, concluding that the limit does not exist for this function at x equals 2.

Takeaways
  • 📈 The function f(x) is piecewise continuous and defined differently over specified intervals.
  • 🤔 For 0 < x ≤ 2, f(x) = ln(x), and for x > 2, f(x) = x^2 * ln(x).
  • 🎯 The goal is to find the limit of f(x) as x approaches 2.
  • 🔍 To determine the limit, consider the left-hand limit (approaching 2 from the left) and the right-hand limit (approaching 2 from the right).
  • 📌 The left-hand limit is ln(2), as it directly applies to the first interval's definition.
  • 📊 The right-hand limit is 4 * ln(2), calculated by applying the second interval's definition at x = 2.
  • 🌟 Both left-hand and right-hand limits exist, but they are not equal.
  • 🚫 The function has a jump discontinuity at x = 2 due to the differing limits.
  • ❌ Since the left-hand and right-hand limits are not the same, the limit of f(x) as x approaches 2 does not exist.
  • 📈 The behavior of the function at x = 2 illustrates the concept of one-sided limits and their importance in understanding the continuity of functions.
Q & A

  • What is the definition of the function f(x) for x values less than or equal to two?

    -For x values less than or equal to two, the function f(x) is defined as the natural log of x.

  • How is the function f(x) defined for x values greater than two?

    -For x values greater than two, f(x) is defined as x squared times the natural log of x.

  • What is the main goal of the script?

    -The main goal of the script is to find the limit of f(x) as x approaches two.

  • What is the significance of the value two in the context of this function?

    -The value two is significant because it serves as the boundary between the two intervals over which the function is defined differently.

  • What is the value of f(x) at x equals two?

    -At x equals two, the value of f(x) is the natural log of two, since two falls under the first interval where x is less than or equal to two.

  • What does it mean for a limit to exist?

    -A limit exists if the left-hand limit and the right-hand limit both exist and are equal to the same value.

  • What is the left-hand limit of f(x) as x approaches two?

    -The left-hand limit of f(x) as x approaches two is the natural log of two, which is the value of the function at two according to the first interval.

  • What is the right-hand limit of f(x) as x approaches two?

    -The right-hand limit of f(x) as x approaches two is four times the natural log of two, which is the result of evaluating the second interval's definition at two.

  • What is the conclusion about the limit of f(x) as x approaches two based on the script?

    -The limit of f(x) as x approaches two does not exist because the left-hand limit and the right-hand limit are not equal, indicating a jump discontinuity at x equals two.

  • What would be observed on the graph of this function at x equals two?

    -On the graph of this function, there would be a jump discontinuity at x equals two, reflecting the different values of the left-hand and right-hand limits.

  • Why are the left-hand and right-hand limits of f(x) different when approaching x equals two?

    -The left-hand and right-hand limits are different because the function is defined by different mathematical expressions on either side of x equals two, leading to a discontinuity at that point.

Outlines
00:00
📚 Analysis of a Piecewise Continuous Function

This paragraph discusses the evaluation of a piecewise continuous function, f(x), defined over different intervals. The function is the natural log of x for values less than or equal to two and becomes x squared times the natural log of x for values greater than two. The main focus is on finding the limit of f(x) as x approaches two, considering the boundary between the two intervals. It is noted that while the function can be evaluated directly at two (natural log of two), the limit requires consideration of the left and right-hand limits. The left-hand limit is the natural log of two, while the right-hand limit is four times the natural log of two, indicating a jump discontinuity at x equals two. Consequently, the limit does not exist due to the discrepancy between the left and right-hand limits.

Mindmap
Keywords
💡Function
In mathematics, a function is a relation that assigns a single output value to each input value. In the context of the video, 'f of x' is a function that is piecewise continuous, meaning it is defined by different expressions for different intervals of 'x'. The function is central to the discussion as the video aims to find the limit of this function as 'x' approaches two.
💡Piecewise Continuous
A piecewise continuous function is one that is continuous on each individual piece or segment of its domain but may have points of discontinuity between these segments. In the video, the function 'f of x' is described as piecewise continuous with different definitions for 'x' values less than or equal to 2 and greater than 2.
💡Limit
In calculus, a limit is a value that a function or sequence 'approaches' as the input (or index) approaches some value. The concept is fundamental to understanding continuity, derivatives, and integrals. In the video, the limit of 'f of x' as 'x' approaches 2 is the main focus, with the exploration of left-hand and right-hand limits to determine if the function has a well-defined limit at 'x = 2'.
💡Natural Logarithm
The natural logarithm, often denoted as 'ln', is the logarithm to the base 'e', where 'e' is the mathematical constant approximately equal to 2.71828. It is a key concept in mathematics and is used in the video to define part of the function 'f of x' for values of 'x' between 0 and 2.
💡Boundary
In the context of the video, a boundary refers to a point or value that separates the domain of a function into different intervals where the function may have different definitions. The boundary is significant because it can affect the continuity and the limit of the function.
💡Left-hand Limit
The left-hand limit of a function at a certain point is the value the function approaches as the input gets arbitrarily close to that point from the left side. It is one of the ways to analyze the behavior of a function near its discontinuities, as demonstrated in the video when discussing the limit of 'f of x' as 'x' approaches 2 from the left.
💡Right-hand Limit
The right-hand limit of a function at a certain point is the value the function approaches as the input gets arbitrarily close to that point from the right side. It is crucial for determining the continuity and existence of limits at points where the function may have different behaviors on either side.
💡Discontinuity
A discontinuity in a function is a point where the function is not continuous; in other words, the function may have a jump, hole, or infinite behavior at that point. The video discusses a jump discontinuity at the boundary value of 2 for the function 'f of x', where the left-hand and right-hand limits are not equal.
💡Well-defined Limit
A well-defined limit is one where the left-hand and right-hand limits of a function at a certain point exist and are equal. This indicates that the function approaches the same value from both sides at that point, and thus the limit is clear and unambiguous.
💡Square
The square of a number is the result of multiplying that number by itself. In the video, the square of 'x' is used in the definition of the function 'f of x' for values of 'x' greater than 2, as part of the expression 'x squared times the natural log of x'.
💡Graph
A graph is a visual representation of the relationship between variables, in this case, the function 'f of x' and its input 'x'. The graph can illustrate the behavior of the function, including points of continuity and discontinuity. The video mentions that if graphed, the function 'f of x' would show a jump at 'x = 2', indicating a discontinuity.
Highlights

The function f(x) is piecewise continuous, defined differently for x < 2 and x > 2.

For 0 < x ≤ 2, f(x) = ln(x), providing a natural logarithmic relationship.

For x > 2, f(x) = x^2 * ln(x), introducing a quadratic factor multiplied by the natural logarithm.

The limit of f(x) as x approaches 2 is of interest, with 2 being the boundary between the two intervals.

Evaluating f(x) at 2 directly yields ln(2) due to the first interval's definition.

To find the limit, consider the left-hand limit (approaching 2 from the left) and the right-hand limit (from the right).

The left-hand limit is ln(2), as it's continuous within the interval 0 < x ≤ 2.

The right-hand limit is 2^2 * ln(2) = 4 * ln(2), as it's continuous for x ≥ 2.

Despite both limits existing, they are not equal, indicating a jump discontinuity at x = 2.

The function's graph would show a jump at x = 2, demonstrating the discontinuity.

The limit of f(x) as x approaches 2 does not exist due to the discrepancy between the left and right limits.

This analysis demonstrates the importance of evaluating one-sided limits for piecewise functions at interval boundaries.

The concept of continuity is crucial in determining the existence of limits for piecewise functions.

The function's behavior at the boundary points can significantly impact its overall continuity and limit properties.

This example illustrates the mathematical process of evaluating limits for piecewise continuous functions.

Understanding the definition of each piece is essential for determining the function's behavior at interval boundaries.

Transcripts

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Related Tags
MathematicsCalculusLimitsPiecewise FunctionsDiscontinuityNatural LogarithmFunction EvaluationLeft-Right LimitsContinuous FunctionsMath Education