Introduction to limits 2 | Limits | Precalculus | Khan Academy

Summary

In 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

00:00

📊 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.

05:03

📉 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.

Keywords

💡Limit

A limit describes the value that a function approaches as the input (x) gets closer to a specific point. In the video, the limit is used to explore how a function behaves near a certain value, even if the function’s actual value at that point might be different or undefined. For example, the limit as x approaches 2 for the function x^2 is 4, meaning as x gets closer to 2, the value of x^2 approaches 4.

💡x approaches 2

This phrase refers to the behavior of a function as the input value (x) gets closer to 2 from both sides (less than 2 and greater than 2). The speaker discusses how x approaching 2 allows us to analyze the value of x^2, which is critical in understanding limits. In the example, as x gets closer to 2, the function approaches a value of 4.

💡f(x)

f(x) represents a function of x, a mathematical expression that gives a corresponding output for each input. In the video, f(x) is defined as x^2 except at x = 2, where it is manually set to 3. This discrepancy in the function's definition at x = 2 is used to show how the concept of limits helps analyze function behavior despite discontinuities.

💡Graph of x squared (x^2)

The graph of x^2 is a visual representation of the function y = x^2. It forms a parabola, and the video uses this graph to explain how the function behaves as x approaches 2. The graph helps illustrate the concept of a limit, as it shows the value the function approaches from both sides of x = 2.

💡Hole in the graph

A hole in the graph refers to a point where a function is not defined, often due to a special condition in the function’s definition. In the example, the function is defined as x^2 except at x = 2, where it equals 3. This creates a ‘hole’ at (2, 4) on the x^2 graph, highlighting the need for limits to analyze the function's behavior near that point.

💡Discontinuity

A discontinuity occurs when a function has an abrupt change or is not defined at a particular point. In the video, the function has a discontinuity at x = 2 because f(x) is defined as x^2 for all values except at 2, where it is set to 3. This example demonstrates how limits help analyze the behavior of functions even at discontinuities.

💡Left-hand limit

The left-hand limit is the value a function approaches as x approaches a particular point from the left, or from values smaller than that point. In the video, the left-hand limit of f(x) as x approaches 2 is 4, because the values of x^2 get closer to 4 as x increases toward 2 from the left side.

💡Right-hand limit

The right-hand limit refers to the value a function approaches as x approaches a certain point from the right, or from values larger than that point. In the example, the right-hand limit of f(x) as x approaches 2 is also 4, because as x decreases toward 2 from the right, the function x^2 approaches 4.

💡Evaluating the function

Evaluating the function means finding the exact value of a function at a particular point by substituting that point into the function’s expression. In the video, f(2) is evaluated as 3, based on the special condition given for f(x). The difference between evaluating the function and finding the limit shows why limits are important when direct evaluation is not straightforward.

💡Delta-epsilon definition

The delta-epsilon definition is the formal, mathematical way of defining limits. It rigorously describes how a function can get arbitrarily close to a certain value (the limit) by ensuring that x is sufficiently close to a target value. The speaker mentions that this definition will be covered in future lessons, indicating its importance for a more precise understanding of limits.

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.