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A short class with Sarah and her students. The other ways are under More.
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Today, we will learn about limits. A limit describes how a function behaves as the input approaches a certain value. Can anyone tell me what this might mean?
Does it mean we are looking at the function’s output near that input value?
Exactly! We write this as lim𝑓(𝑥) when 𝑥 approaches a. If 𝑓(𝑥) gets nearer to some number L, we say that the limit of 𝑓(𝑥) as 𝑥 approaches 𝑎 is L.
So, it's not always defined at that point?
Right! And that brings us to evaluating limits. For example, if 𝑓(𝑥) = 𝑥² + 3, what do you think lim as 𝑥 approaches 2 would be?
I think it would be 7?
Correct! That's how we evaluate limits.
Now, let's explore how to evaluate limits using a table. If we have the function 𝑓(𝑥) = 𝑥² + 3, we can create a table of values as 𝑥 approaches 2.
Do we just choose values around 2?
Exactly! We can look at values like 1.9, 2.0, and 2.1. Who can tell me what we get?
We get outputs that approach 7, right?
Great! So we can conclude that lim as 𝑥 approaches 2 of 𝑓(𝑥) equals 7.
Next, let's discuss how we can evaluate limits graphically. Imagine you are looking at the graph of a function. As you get closer to a point from both sides, what do you observe?
We see where the graph levels off, indicating a limit.
What if the graph approaches two different values?
Good question! If it approaches different values, then the limit does not exist at that point. Can you think of an example of such behavior?
Maybe a jump discontinuity?
Exactly! Well done.
Sometimes we can evaluate limits algebraically. Using direct substitution can be effective. For example, how would we evaluate lim as 𝑥 approaches 2 of 3𝑥 + 1?
We substitute 2 into the function and get 7.
Exactly! But what happens if we have an indeterminate form like 0/0? Let’s try lim as 𝑥 approaches 1 of (𝑥² - 1)/(𝑥 - 1).
We would need to factor it first.
Correct! After factoring, what do we find?
Then it simplifies to 2?
Exactly! Great insights.
Lastly, we need to cover one-sided limits. What do we mean when we approach from the left or right?
That we focus only on one side of the point?
Exactly! Can anyone give an example of when a limit doesn’t exist due to different left-hand and right-hand limits?
Like if one is 2 and the other side is 4?
Correct! Now, what if a function increases without bound? What do we call those limits?
Infinite limits!
Great job, everyone! Remember that understanding limits is essential as we move into calculus.
The value a function approaches as the input approaches a specific value.
Plugging a value directly into the function to evaluate the limit.
A form that indicates the need for further analysis, such as 0/0.
Limits that consider only one side of the approach.
A limit that suggests the function goes to infinity.
When the left and right limits do not match.
When approaching values, take your stance, a limit shows where functions dance!
Imagine a river flowing to a lake; it’s the point of limits where functions no longer break. It leads us to the values that we seek as the flow builds suspense, avoiding the bleak.
LIM - Limit Involves Movement (LIM) highlights that a limit tracks how a function moves/behaves as input approaches a point.
L.I.M - Limit Illustrates Mapping the analogy refers to a ‘jump discontinuity’ in a graph, where the function has different outputs from each side
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What is a limit?
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What is direct substitution?
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What denotes an indeterminate form?
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What is an infinite limit?
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The value that a function approaches as the input approaches a specified point.
The method of substituting the value directly into the function to find limits.
A form that arises in limits where direct substitution does not yield a clear result, such as 0/0.
Evaluating a limit approaching a point from one side only, either left or right.
A limit where the function increases or decreases without bound as it approaches a certain point.
A conclusion reached when a function approaches different values from the left and right or oscillates infinitely near the point.
This section introduces the concept of limits in calculus, explaining how to evaluate limits numerically and graphically.
Understanding limits is fundamental in calculus, serving as the bedrock for differentiation and integration. This section explains what limits are, how to estimate them from tables and graphs, and introduces key types such as one-sided and infinite limits.
In calculus, a limit is defined as the value that a function approaches as the input approaches a specified point. This concept is crucial because it allows us to understand the behavior of functions in a more nuanced manner, especially at points where they may not be explicitly defined. In this section, we will explore different methods of evaluating limits, including numerical and graphical approaches, direct substitution, and simplifications needed for indeterminate forms. We will also discuss one-sided limits and situations when a limit does not exist, enhancing our comprehension of function behavior near critical points.