Introduction to limits 2 | Limits | Precalculus | Khan Academy
Summary
TLDRIn this video, the concept of limits in calculus is introduced. The speaker explains how limits determine the value that a function approaches as the input approaches a specific point. Using the example of the function f(x) = x², the speaker demonstrates how limits work, even when the function is not directly defined at that point. The key takeaway is that limits can provide insight into the behavior of functions near points where they may not be explicitly defined, setting the stage for deeper mathematical concepts like derivatives and integrals.
Takeaways
- 📝 The presentation is an introduction to the concept of limits in calculus.
- 🟢 The limit is written as 'lim' followed by the expression and x approaching a specific value.
- 🔍 The example used is the limit of x^2 as x approaches 2, which equals 4.
- 📊 A graph of x^2 helps illustrate how the function behaves as x approaches 2.
- 🕳 A 'gap' in the function occurs when a specific value, like x=2, is defined differently, showing the purpose of limits.
- 📉 In the modified function, f(x) equals x^2 when x ≠ 2 and equals 3 when x = 2.
- 🔢 The limit still approaches 4 as x approaches 2, even though f(2) is defined as 3.
- ❓ Limits help explain function behavior when functions have discontinuities or aren't defined at certain points.
- 🤔 The key difference between limits and evaluating functions at a specific point is emphasized.
- 🎓 A more formal mathematical definition of limits (delta-epsilon) will be covered later, along with additional problem-solving.
Q & A
What is the main concept being introduced in this presentation?
-The main concept being introduced is the limit of a function, which examines what value an expression approaches as the input variable gets closer to a certain point.
How is the limit of x² as x approaches 2 calculated?
-The limit of x² as x approaches 2 is calculated by observing the behavior of the function as x gets closer to 2 from both sides. As x approaches 2, the value of x² approaches 4.
Why might someone find the concept of a limit unnecessary at first?
-At first, the concept of a limit may seem unnecessary because you can often plug in the value directly into the function, as in the case of x², where f(2) = 4. This gives the same result as finding the limit.
What variation does the speaker introduce to explain the importance of limits?
-The speaker introduces a variation where the function f(x) is defined as x² for all values except x = 2, and at x = 2, f(x) is defined to be 3 instead of 4.
What is the limit of f(x) in the modified example as x approaches 2?
-The limit of f(x) as x approaches 2 is still 4, because from both sides of 2, the function behaves like x² and approaches 4, even though f(2) is defined to be 3.
How does the graph change in the modified example where f(x) equals 3 when x equals 2?
-In the modified example, the graph of f(x) looks like the curve of x² but with a gap at x = 2. Instead of continuing smoothly, there is a hole at (2, 4), and the point (2, 3) is marked below the hole.
Why is the limit concept important in the modified example?
-The limit concept is important because it shows that even though the function value at x = 2 is different (f(2) = 3), the values around x = 2 still approach 4. This highlights that limits describe the behavior of a function near a point, even if the function is not defined or jumps at that point.
What does the limit concept reveal about the relationship between f(2) and the limit as x approaches 2?
-The limit concept reveals that f(2) does not always have to equal the limit as x approaches 2. In the modified example, the limit is 4, but f(2) is defined as 3, showing that they can be different.
What does the speaker say will be introduced in future presentations?
-In future presentations, the speaker plans to introduce the formal mathematical definition of a limit using the delta-epsilon method and to work through various problems to build intuition about limits.
Why are limits important in calculus, according to the speaker?
-Limits are important in calculus because they provide the foundation for understanding derivatives and integrals. The concept of a limit allows for precise definitions of instantaneous rates of change and areas under curves, which are essential in calculus.
Outlines
📊 Introduction to Limits and Initial Problem Setup
In this section, the speaker begins the presentation by introducing the concept of limits. They describe the limit as x approaches 2 for the expression x², explaining that it essentially evaluates to 4. To illustrate, a rough graph of the function x² is drawn. The speaker explains how limits describe the behavior of a function as x approaches a particular value from both sides. Although this seems straightforward for continuous functions, the introduction hints at more complex cases where the function is not defined at certain points.
📉 Introducing Discontinuous Functions and Their Limits
The speaker introduces a modified version of the function f(x), where f(x) = x² for all x except when x = 2, where f(x) is defined as 3. This creates a discontinuity or a 'gap' at x = 2, represented visually with a hole in the graph. The speaker then revisits the limit as x approaches 2, explaining that although the function jumps to a different value at x = 2, the limit as x approaches from both sides still equals 4. This section highlights the distinction between the limit of a function and the value of the function at a specific point.
Mindmap
Keywords
💡Limit
💡x approaches 2
💡f(x)
💡Graph of x squared (x^2)
💡Hole in the graph
💡Discontinuity
💡Left-hand limit
💡Right-hand limit
💡Evaluating the function
💡Delta-epsilon definition
Highlights
Introduction to limits and the concept of approaching a value.
Explanation of how to write limits using notation: 'limit as x approaches a value'.
Graphical representation of the function x squared and how it relates to limits.
As x approaches 2 from both sides, the expression x squared approaches 4.
Introduction of a new concept: a function that behaves differently at a specific point.
Definition of the function f(x), which equals x squared except at x=2, where f(x)=3.
Graphical explanation of a function with a gap at x=2 and f(2)=3.
Limit of f(x) as x approaches 2 still equals 4, even though f(2)=3.
Key insight: limits describe the behavior of a function as it approaches a point, even if the function’s value at that point is different.
Importance of limits when the function is not defined or has a discontinuity at a specific point.
Limits as a broader concept for understanding function behavior near specific points.
Difference between calculating a limit and simply evaluating a function at a given point.
Introduction to the idea that limits are foundational for understanding derivatives and integrals.
Hints at a more formal definition of limits using delta-epsilon notation.
Next steps: solving problems involving limits to build intuition and understanding.
Transcripts
Welcome to the presentation on limits.
Let's get started with some-- well, first an explanation
before I do any problems.
So let's say I had-- let me make sure I have the right
color and my pen works.
OK, let's say I had the limit, and I'll explain what a
limit is in a second.
But the way you write it is you say the limit-- oh, my color is
on the wrong-- OK, let me use the pen and yellow.
OK, the limit as x approaches 2 of x squared.
Now, all this is saying is what value does the expression x
squared approach as x approaches 2?
Well, this is pretty easy.
If we look at-- let me at least draw a graph.
I'll stay in this yellow color.
So let me draw.
x squared looks something like-- let me use
a different color.
x square looks something like this, right?
And when x is equal to 2, y, or the expression-- because
we don't say what this is equal to.
It's just the expression-- x squared is equal to 4, right?
So a limit is saying, as x approaches 2, as x approaches 2
from both sides, from numbers left than 2 and from numbers
right than 2, what does the expression approach?
And you might, I think, already see where this is going and be
wondering why we're even going to the trouble of learning this
new concept because it seems pretty obvious, but as x-- as
we get to x closer and closer to 2 from this direction, and
as we get to x closer and closer to 2 to this
direction, what does this expression equal?
Well, it essentially equals 4, right?
The expression is equal to 4.
The way I think about it is as you move on the curve closer
and closer to the expression's value, what does the
expression equal?
In this case, it equals 4.
You're probably saying, Sal, this seems like a useless
concept because I could have just stuck 2 in there, and I
know that if this is-- say this is f of x, that if f of x is
equal to x squared, that f of 2 is equal to 4, and that would
have been a no-brainer.
Well, let me maybe give you one wrinkle on that, and hopefully
now you'll start to see what the use of a limit is.
Let me to define-- let me say f of x is equal to x squared
when, if x does not equal 2, and let's say it equals
3 when x equals 2.
Interesting.
So it's a slight variation on this expression right here.
So this is our new f of x.
So let me ask you a question.
What is-- my pen still works-- what is the limit-- I used
cursive this time-- what is the limit as x-- that's an x--
as x approaches 2 of f of x?
That's an x.
It says x approaches 2.
It's just like that.
I just-- I don't know.
For some reason, my brain is working functionally.
OK, so let me graph this now.
So that's an equally neat-looking graph as
the one I just drew.
Let me draw.
So now it's almost the same as this curve, except something
interesting happens at x equals 2.
So it's just like this.
It's like an x squared curve like that.
But at x equals 2 and f of x equals 4, we
draw a little hole.
We draw a hole because it's not defined at x equals 2.
This is x equals 2.
This is 2.
This is 4.
This is the f of x axis, of course.
And when x is equal to 2-- let's say this is 3.
When x is equal to 2, f of x is equal to 3.
This is actually right below this.
I should-- it doesn't look completely right below it,
but I think you got to get the picture.
See, this graph is x squared.
It's exactly x squared until we get to x equals 2.
At x equals 2, We have a grap-- No, not a grap.
We have a gap in the graph, which maybe
could be called a grap.
We have a gap in the graph, and then we keep-- and then after x
equals 2, we keep moving on.
And that gap, and that gap is defined right here, what
happens when x equals 2?
Well, then f of x is equal to 3.
So this graph kind of goes-- it's just like x squared, but
instead of f of 2 being 4, f of 2 drops down to 3, but
then we keep on going.
So going back to the limit problem, what is the
limit as x approaches 2?
Now, well, let's think about the same thing.
We're going to go-- this is how I visualize it.
I go along the curve.
Let me pick a different color.
So as x approaches 2 from this side, from the left-hand side
or from numbers less than 2, f of x is approaching values
approaching 4, right? f of x is approaching 4 as x
approaches 2, right?
I think you see that.
If you just follow along the curve, as you approach f of 2,
you get closer and closer to 4.
Similarly, as you go from the right-hand side-- make sure
my thing's still working.
As you go from the right-hand side, you go along the
curve, and f of x is also slowly approaching 4.
So, as you can see, as we go closer and closer and
closer to x equals 2, f of whatever number that is
approaches 4, right?
So, in this case, the limit as x approaches
2 is also equal to 4.
Well, this is interesting because, in this case, the
limit as x approaches 2 of f of x does not equal f of 2.
Now, normally, this would be on this line.
In this case, the limit as you approach the expression is
equal to evaluating the expression of that value.
In this case, the limit isn't.
I think now you're starting to see why the limit is a slightly
different concept than just evaluating the function at
that point because you have functions where, for whatever
reason at a certain point, either the function might not
be defined or the function kind of jumps up or down, but as you
approach that point, you still approach a value different than
the function at that point.
Now, that's my introduction.
I think this will give you intuition for what a limit is.
In another presentation, I'll give you the more formal
mathematical, you know, the delta-epsilon
definition of a limit.
And actually, in the very next module, I'm now going to
do a bunch of problems involving the limit.
I think as you do more and more problems, you'll get more and
more of an intuition as to what a limit is.
And then as we go into drill derivatives and integrals,
you'll actually understand why people probably even invented
limits to begin with.
We'll see you in the next presentation.
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