Limit at a point of discontinuity
TLDRThe video script discusses the limit of a function, f(x), as x approaches three, where the function is undefined at x equals three. By rewriting the function for x greater than and less than three, it's shown that the function simplifies to 1 for x > 3 and -1 for x < 3. The limit from the left (negative direction) approaches -1, while from the right (positive direction) it approaches 1. The inconsistency in these limits indicates that the overall limit does not exist, as the function does not approach the same value from both directions.
Takeaways
- 🌟 The original function f(x) is undefined at x=3 due to a zero over zero indeterminate form.
- 📝 To address the issue, the function is rewritten as a piecewise function with separate expressions for x > 3 and x < 3.
- 🔢 For x > 3, the function simplifies to f(x) = 1, as the numerator and denominator are the same and the absolute value has no effect.
- 🔽 For x < 3, after applying the absolute value, the function simplifies to f(x) = -1, as the numerator is negative and the denominator remains the same.
- 📈 The function's behavior is visualized on a graph with x on the horizontal axis and f(x) on the vertical axis.
- 🏁 At x=3, the function is undefined, and the graph has a vertical asymptote.
- 🠔 When approaching x=3 from the left (negative direction), the function approaches -1.
- 🠕 When approaching x=3 from the right (positive direction), the function approaches 1.
- 🔄 The limit from the left is different from the limit from the right, indicating that the overall limit does not exist for x approaching 3.
- 📌 The existence of a function's limit at a point requires that the limits from both the left and right directions are equal.
- 🚫 Since the limits from both sides are not equal in this case (-1 from the left and 1 from the right), the function has no limit as x approaches 3.
Q & A
What is the original function given in the transcript?
-The original function given is f(x) = |x - 3| / (x - 3).
Why is the function not defined at x = 3?
-The function is not defined at x = 3 because it results in a zero over zero indeterminate form.
How does the speaker attempt to address the indeterminate form at x = 3?
-The speaker rewrites the function as a piecewise function, considering two cases: when x > 3 and when x < 3.
What is the value of f(x) for x > 3?
-For x > 3, the value of f(x) is 1, since the numerator and denominator are the same and the absolute value does not change the value.
What is the value of f(x) for x < 3?
-For x < 3, the value of f(x) is -1, as the numerator becomes negative and the absolute value changes the sign, resulting in a negative one over a positive denominator.
How does the speaker visualize the function on a graph?
-The speaker visualizes the function by plotting the y-axis as f(x) and marking values for x greater and less than 3, showing that f(x) approaches 1 from the right and -1 from the left, but is undefined at x = 3.
What is the limit of f(x) as x approaches 3 from the left?
-The limit of f(x) as x approaches 3 from the left is -1, as the function consistently approaches -1 for values of x just less than 3.
What is the limit of f(x) as x approaches 3 from the right?
-The limit of f(x) as x approaches 3 from the right is 1, as the function consistently approaches 1 for values of x just greater than 3.
Does the limit of f(x) exist as x approaches 3?
-The limit of f(x) does not exist as x approaches 3 because the function approaches different values from the left (-1) and from the right (1).
What is the condition for a limit to exist?
-A limit exists if and only if the limit from the left and the limit from the right both exist and are equal to the same value L.
What conclusion does the speaker reach about the function based on the limits?
-The speaker concludes that the function does not have a limit at x = 3 because the limits from the left and right are not equal.
What advice does the speaker give to verify the conclusions?
-The speaker advises to try out the function with different values of x, both greater and less than 3, to observe the behavior of the function and confirm the conclusions.
Outlines
📚 Analysis of a Function's Limit
The paragraph discusses the limit of a function, f(x), as x approaches three. Initially, the function is undefined at x=3 due to a zero over zero indeterminate form. To address this, the function is rewritten for two cases: when x > 3 and x < 3. For x > 3, the function simplifies to 1, and for x < 3, it simplifies to -1. This is verified by testing values close to 3 from both directions. The visual representation of the function on a graph is also described, with the function being undefined at x=3 but equal to 1 for x > 3 and -1 for x < 3. The paragraph concludes by examining the limit from both the left (negative direction) and the right (positive direction), noting that the function approaches -1 and 1, respectively, and thus the limit does not exist as the left and right limits are not equal.
🔢 Understanding the Non-Existence of a Limit
This paragraph delves into the concept of the limit not existing for the given function at x=3. It explains that a limit exists if and only if the limit from the left (negative direction) is equal to the limit from the right (positive direction). However, in this case, the function approaches -1 from the left and +1 from the right, indicating that the limits are not equal. Therefore, the overall limit as x approaches three does not exist. The explanation is clear and emphasizes the importance of the function's behavior from both directions in determining the existence of a limit.
Mindmap
Keywords
💡limit
💡absolute value
💡undefined function
💡piecewise function
💡indeterminate form
💡one over x
💡negative direction
💡positive direction
💡approaching
💡function behavior
💡limits from both sides
Highlights
The function f(x) is defined as the absolute value of (x - 3) over (x - 3), which is not defined at x equals 3 due to a zero over zero indeterminate form.
To find the limit of f(x) as x approaches 3, the function is rewritten piecewise for x greater than 3 and x less than 3.
For x greater than 3, the function simplifies to 1, since the numerator and denominator are the same and the absolute value does not change the value.
For x less than 3, the function simplifies to -1, as the numerator becomes negative and the absolute value flips the sign.
By testing values not equal to 3, it is confirmed that the function equals 1 for any x greater than 3 and -1 for any x less than 3.
A visual representation of the function is provided by plotting it on a graph with x equal to 3 being the point of interest.
The limit of the function as x approaches 3 from the negative direction (left-hand limit) is -1, as demonstrated by approaching values just below 3.
The limit of the function as x approaches 3 from the positive direction (right-hand limit) is 1, as shown by values just above 3.
The limit does not exist as x approaches 3 because the left-hand limit and the right-hand limit are not equal, with the left-hand limit being -1 and the right-hand limit being 1.
The definition of a limit being equal to L is that the limit from both the negative and positive directions must also equal L; this condition is not met in this case.
The concept of the function being undefined at x equals 3 is explained through the concept of zero over zero indeterminate form.
The piecewise function definition helps in understanding the behavior of the function for different ranges of x values.
The absolute value function plays a crucial role in determining the sign of the function based on the value of x.
The use of specific test values, such as 3.1, 3.001, 3.5, 4, and 7, provides evidence for the behavior of the function as x approaches 3.
The concept of approaching a point from the left and right is introduced to analyze the one-sided limits of the function.
The absence of a common limit from both directions indicates that the overall limit as x approaches 3 does not exist for the given function.
The explanation emphasizes the importance of consistent limits from both directions for the existence of a function's limit at a particular point.
Transcripts
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