# Introduction To Limits

## Introduction To Limits Meine Schüler

Introduction to limits: easy mathematics | Adrian Harrison | ISBN: | Kostenloser Versand für alle Bücher mit Versand und Verkauf duch Amazon. Introduction to limits of functions. Didaktischer Vortrag | - Gong Chen (University of Toronto). I will informally introduce the idea of limits. Trigonometrie · Mathematik Bücher · Lernen · Studio. Limits (An Introduction) Algebra, Tägliches Mathematik, Trigonometrie, Mathematik Bücher, Lernen. Teaching resource | Unbounded Limit - The limit of the graph approaches infinity.​, Equal one-sided limits - The limit exists. Kostenloser Matheproblemlöser beantwortet Fragen zu deinen Hausaufgaben in Algebra, Geometrie, Trigonometrie, Analysis und Statistik mit. ​Provides a quick introduction to the subject of inverse limits with set-valued function Contains numerous examples and models of the inverse limits Several of. Calculus 1 - Introduction to Limits von The Organic Chemistry Tutor vor 2 Jahren 40 Minuten Aufrufe This calculus 1 review provides. introduction of a borrowing limit or leverage ratio is being considered. In this respect the limits to leverage could for example either consist in a threshold that.

While our question is not precisely formed what constitutes "near the value 1''? By considering Figure 1.

We have. Such an expression gives no information about what is going on with the function nearby. Before continuing, it will be useful to establish some notation.

We write all this as. This is not a complete definition that will come in the next section ; this is a pseudo-definition that will allow us to explore the idea of a limit.

Once we have the true definition of a limit, we will find limits analytically ; that is, exactly using a variety of mathematical tools.

For now, we will approximate limits both graphically and numerically. Graphing a function can provide a good approximation, though often not very precise.

Numerical methods can provide a more accurate approximation. We have already approximated limits graphically, so we now turn our attention to numerical approximations.

This is done in Figure 1. We already approximated the value of this limit as 1 graphically in Figure 1. The table in Figure 1. This is done in Figures 1.

The graph and the table imply that. This example may bring up a few questions about approximating limits and the nature of limits themselves.

Graphs are useful since they give a visual understanding concerning the behavior of a function. Since graphing utilities are very accessible, it makes sense to make proper use of them.

Since tables and graphs are used only to approximate the value of a limit, there is not a firm answer to how many data points are "enough. In Example 1, we used both values less than and greater than 3.

While this is not far off, we could do better. Using values "on both sides of 3'' helps us identify trends. Note that this is a piecewise defined function, so it behaves differently on either side of 0.

Figure 1. We don't know what this function equals at 1. We never defined it. This definition of the function doesn't tell us what to do with 1.

It's literally undefined, literally undefined when x is equal to 1. So this is the function right over here. And so once again, if someone were to ask you what is f of 1, you go, and let's say that even though this was a function definition, you'd go, OK x is equal to 1, oh wait there's a gap in my function over here.

It is undefined. So let me write it again. It's kind of redundant, but I'll rewrite it f of 1 is undefined.

But what if I were to ask you, what is the function approaching as x equals 1. And now this is starting to touch on the idea of a limit.

So as x gets closer and closer to 1. So as we get closer and closer x is to 1, what is the function approaching. Well, this entire time, the function, what's a getting closer and closer to.

On the left hand side, no matter how close you get to 1, as long as you're not at 1, you're actually at f of x is equal to 1. Over here from the right hand side, you get the same thing.

So you could say, and we'll get more and more familiar with this idea as we do more examples, that the limit as x and L-I-M, short for limit, as x approaches 1 of f of x is equal to, as we get closer, we can get unbelievably, we can get infinitely close to 1, as long as we're not at 1.

And our function is going to be equal to 1, it's getting closer and closer and closer to 1. It's actually at 1 the entire time.

So in this case, we could say the limit as x approaches 1 of f of x is 1. So once again, it has very fancy notation, but it's just saying, look what is a function approaching as x gets closer and closer to 1.

Let me do another example where we're dealing with a curve, just so that you have the general idea. So let's say that I have the function f of x, let me just for the sake of variety, let me call it g of x.

Let's say that we have g of x is equal to, I could define it this way, we could define it as x squared, when x does not equal, I don't know when x does not equal 2.

And let's say that when x equals 2 it is equal to 1. So once again, a kind of an interesting function that, as you'll see, is not fully continuous, it has a discontinuity.

Let me graph it. So this is my y equals f of x axis, this is my x-axis right over here. Let me draw x equals 2, x, let's say this is x equals 1, this is x equals 2, this is negative 1, this is negative 2.

And then let me draw, so everywhere except x equals 2, it's equal to x squared. So let me draw it like this. So it's going to be a parabola, looks something like this, let me draw a better version of the parabola.

So it'll look something like this. Not the most beautifully drawn parabola in the history of drawing parabolas, but I think it'll give you the idea.

I think you know what a parabola looks like, hopefully. It should be symmetric, let me redraw it because that's kind of ugly.

And that's looking better. OK, all right, there you go. All right, now, this would be the graph of just x squared.

But this can't be. It's not x squared when x is equal to 2. So once again, when x is equal to 2, we should have a little bit of a discontinuity here.

So I'll draw a gap right over there, because when x equals 2 the function is equal to 1. When x is equal to 2, so let's say that, and I'm not doing them on the same scale, but let's say that.

So this, on the graph of f of x is equal to x squared, this would be 4, this would be 2, this would be 1, this would be 3.

So when x is equal to 2, our function is equal to 1. So this is a bit of a bizarre function, but we can define it this way.

You can define a function however you like to define it. And so notice, it's just like the graph of f of x is equal to x squared, except when you get to 2, it has this gap, because you don't use the f of x is equal to x squared when x is equal to 2.

You use f of x-- or I should say g of x-- you use g of x is equal to 1. Have I been saying f of x? I apologize for that.

You use g of x is equal to 1. So then then at 2, just at 2, just exactly at 2, it drops down to 1. And then it keeps going along the function g of x is equal to, or I should say, along the function x squared.

So my question to you. So there's a couple of things, if I were to just evaluate the function g of 2. Well, you'd look at this definition, OK, when x equals 2, I use this situation right over here.

And it tells me, it's going to be equal to 1. Let me ask a more interesting question. Or perhaps a more interesting question. What is the limit as x approaches 2 of g of x.

Once again, fancy notation, but it's asking something pretty, pretty, pretty simple. It's saying as x gets closer and closer to 2, as you get closer and closer, and this isn't a rigorous definition, we'll do that in future videos.

As x gets closer and closer to 2, what is g of x approaching?

This example may bring up a few questions about approximating limits and the nature of limits themselves. Graphs are useful since they give a visual understanding concerning the behavior of a function.

Since graphing utilities are very accessible, it makes sense to make proper use of them. Since tables and graphs are used only to approximate the value of a limit, there is not a firm answer to how many data points are "enough.

In Example 1, we used both values less than and greater than 3. While this is not far off, we could do better.

Using values "on both sides of 3'' helps us identify trends. Note that this is a piecewise defined function, so it behaves differently on either side of 0.

Figure 1. The table shown in Figure 1. There are three ways in which a limit may fail to exist. Recognizing this behavior is important; we'll study this in greater depth later.

We can deduce this on our own, without the aid of the graph and table. However, Figure 1. Here the oscillation is even more pronounced.

Finally, in the table in Figure 1. Because of this oscillation,. I think you know what a parabola looks like, hopefully.

It should be symmetric, let me redraw it because that's kind of ugly. And that's looking better. OK, all right, there you go. All right, now, this would be the graph of just x squared.

But this can't be. It's not x squared when x is equal to 2. So once again, when x is equal to 2, we should have a little bit of a discontinuity here.

So I'll draw a gap right over there, because when x equals 2 the function is equal to 1. When x is equal to 2, so let's say that, and I'm not doing them on the same scale, but let's say that.

So this, on the graph of f of x is equal to x squared, this would be 4, this would be 2, this would be 1, this would be 3. So when x is equal to 2, our function is equal to 1.

So this is a bit of a bizarre function, but we can define it this way. You can define a function however you like to define it.

And so notice, it's just like the graph of f of x is equal to x squared, except when you get to 2, it has this gap, because you don't use the f of x is equal to x squared when x is equal to 2.

You use f of x-- or I should say g of x-- you use g of x is equal to 1. Have I been saying f of x? I apologize for that. You use g of x is equal to 1.

So then then at 2, just at 2, just exactly at 2, it drops down to 1. And then it keeps going along the function g of x is equal to, or I should say, along the function x squared.

So my question to you. So there's a couple of things, if I were to just evaluate the function g of 2. Well, you'd look at this definition, OK, when x equals 2, I use this situation right over here.

And it tells me, it's going to be equal to 1. Let me ask a more interesting question. Or perhaps a more interesting question. What is the limit as x approaches 2 of g of x.

Once again, fancy notation, but it's asking something pretty, pretty, pretty simple. It's saying as x gets closer and closer to 2, as you get closer and closer, and this isn't a rigorous definition, we'll do that in future videos.

As x gets closer and closer to 2, what is g of x approaching? So if you get to 1. Or if you were to go from the positive direction. If you were to say 2.

And you can see it visually just by drawing the graph. As g gets closer and closer to 2, and if we were to follow along the graph, we see that we are approaching 4.

Even though that's not where the function is, the function drops down to 1. The limit of g of x as x approaches 2 is equal to 4.

And you could even do this numerically using a calculator, and let me do that, because I think that will be interesting. So let me get the calculator out, let me get my trusty TI out.

So here is my calculator, and you could numerically say, OK, what's it going to approach as you approach x equals 2. So let's try 1.

So you'd have 1. Differentiability of Basic Functions. Examples on Evaluating Limits Set Introduction To Continuity. Rules Of Differentiation.

Introduction To Differentiability. Examples On Differentiation Set Introduction To Differentiation. Parametric and Implicit Functions. L-Hospital Rule.

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## Introduction To Limits Video

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## Introduction To Limits Video

Limits \u0026 Continuity (1 of 3: Formal intro to limits)

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