One-Sided Limits
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Introduction to One-Sided Limits
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Today we are discussing one-sided limits, which help us understand how a function behaves as it approaches a certain value from one specific side. Let's start with the left-hand limit, which we denote as lim x→a⁻ f(x). This is the limit as x approaches a from the left. Can anyone explain what that means?
It means we look at values of x that are slightly less than a to see what f(x) approaches!
Exactly! Now, can someone define the right-hand limit for me?
It’s lim x→a⁺ f(x), and it looks at values of x that are slightly more than a!
Well done! Understanding these two concepts is crucial for determining if the overall limit exists.
Examples of One-Sided Limits
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Let’s look at an example. Suppose we have lim x→3⁻ f(x) = 2 and lim x→3⁺ f(x) = 4. What can we say about lim x→3 f(x)?
The limit does not exist because the left-hand and right-hand limits are not equal!
Correct! This is crucial for understanding discontinuities in functions. Who can tell me why we care about one-sided limits in calculus?
Because they help us understand what happens around points of discontinuity!
Understanding When Limits Do Not Exist
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Let’s talk about when a limit does not exist. We’ve already covered cases where the left and right limits differ. What else could cause a limit to not exist?
If the function oscillates or goes to infinity near that point?
Exactly! If a function oscillates infinitely or approaches infinity, we also say the limit does not exist. Anyone has an example of a function that behaves like this?
What about f(x) = 1/x as x approaches 0? It goes to positive or negative infinity!
Great example! Understanding these concepts tends to lay the groundwork for further studies in calculus.
Introduction & Overview
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Quick Overview
Standard
One-sided limits are crucial in understanding the behavior of functions near points of discontinuity or undefined values. This section distinguishes between left-hand limits and right-hand limits, providing methods to determine when a limit does not exist due to differences in the limiting values approached from either side.
Detailed
One-Sided Limits
One-sided limits are essential for analyzing the behavior of a function as it approaches a specific point from one direction — either the left or the right. They are defined as follows:
- Left-hand limit: This is denoted as \( \lim_{x\to a^-} f(x) \), which refers to the value that \( f(x) \) approaches as \( x \) approaches \( a \) from the left.
- Right-hand limit: Similarly, \( \lim_{x\to a^+} f(x) \) refers to the value that \( f(x) \) approaches as \( x \) approaches \( a \) from the right.
The concept is particularly useful when the limit may not exist if the left-hand and right-hand limits do not match. For example, if we find:
- \( \lim_{x\to 3^-} f(x) = 2 \)
- \( \lim_{x\to 3^+} f(x) = 4 \)
This indicates that the overall limit \( \lim_{x\to 3} f(x) \) does not exist because the values from either side are different. Understanding one-sided limits helps in analyzing discontinuities and is foundational for concepts like continuity and differentiability in calculus.
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Definition of One-Sided Limits
Chapter 1 of 2
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Chapter Content
A one-sided limit looks at values approaching from only one side.
- Left-hand limit: lim 𝑥→𝑎−𝑓(𝑥)
- Right-hand limit: lim 𝑥→𝑎+𝑓(𝑥)
Detailed Explanation
One-sided limits are used to evaluate the behavior of a function as it approaches a certain point either from the left or the right. The left-hand limit indicates what value the function approaches as the input approaches from values less than a certain point (denoted as x → a⁻). Conversely, the right-hand limit indicates what the function approaches as it moves in from values greater than that point (denoted as x → a⁺). By observing these two limits, we can gather important information about the function’s behavior at that particular point.
Examples & Analogies
Think about a person approaching a door. If they are coming from the left side, we can think of that as the left-hand limit. They might see the door getting closer and closer as they steps forward. Similarly, if a person comes from the right side, we can picture that as the right-hand limit. The door might seem different as they approach from that angle. In mathematics, just like you can assess what you see from each side of the door (the door could look locked from one side and slightly open from the other), one-sided limits allow us to evaluate how the function behaves from each direction as we approach a particular input value.
Example of One-Sided Limits
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Chapter Content
Example:
If:
- lim 𝑥→3−𝑓(𝑥) = 2
- lim 𝑥→3+𝑓(𝑥) = 4
Then:
lim𝑓(𝑥) does not exist
𝑥→3
Because the left and right limits are not equal.
Detailed Explanation
In this example, we are analyzing the function’s limits as x approaches 3 from the left and the right. The left-hand limit tells us that when we approach x = 3 from the left side, the function value gets closer to 2. On the other hand, when we approach from the right side, the function value heads towards 4. Since the two limits (2 from the left and 4 from the right) are not the same, we conclude that the overall limit of the function at x = 3 does not exist (DNE). This discrepancy indicates that there might be a jump or discontinuity at that point.
Examples & Analogies
Imagine standing near a riverbank where there is a bridge. If you walk towards the bridge from the left side, you might see it at a certain height (let’s say 2 meters above the water). However, if you walk towards the bridge from the right side, you notice the bridge is at a different height (4 meters above the water). When you reach the middle of the bridge, you realize that the bridge has shifted heights! Just like the height difference seen in this scenario, the function has different values as it approaches from the left and the right, leading us to say the limit 'does not exist' at that point.
Key Concepts
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One-Sided Limit: The value of the function as it approaches a point from the left or right.
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Left-Hand Limit: Represents the limit as you approach a point from the left, denoted by lim x→a⁻ f(x).
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Right-Hand Limit: Represents the limit as you approach a point from the right, denoted by lim x→a⁺ f(x).
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Limit Does Not Exist (DNE): A limit that is undefined because the left and right limits are not equal.
Examples & Applications
If lim x→2⁻ f(x) = 3 and lim x→2⁺ f(x) = 5, then lim x→2 f(x) does not exist.
The limits of f(x) = 1/(x-1) as x approaches 1 from the left is -∞ and from the right is +∞, thus limit does not exist.
Memory Aids
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Rhymes
Left to the left, right to the right, one-sided limits help us see the light.
Stories
Imagine two friends walking towards a candy store from opposite sides: one approaches from the park, the other from the street. They each see the store from their vantage points, but if they can’t agree on what they see, a limit cannot be established!
Memory Tools
Remember L for Left-hand limit and R for Right-hand limit. They guide you to your limits!
Acronyms
LRL
Limit Right
Limit Left to help you remember to check both sides!
Flash Cards
Glossary
- OneSided Limit
The limit of a function as it approaches a specific point from one side, either left or right.
- LeftHand Limit
The value that a function approaches as the input approaches a certain point from the left side.
- RightHand Limit
The value that a function approaches as the input approaches a certain point from the right side.
- Limit Does Not Exist (DNE)
A situation where a function does not approach the same value from the left and right sides or behaves erratically.
- Infinite Limit
A limit that approaches infinity or negative infinity as it approaches a certain point.
Reference links
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