>> One of the classic problems is called the tangent line problem and the tangent line problem is this. That, you know, in algebra we talk about slope and we can talk about the slope of a line. That's pretty easy. And that slope of the line corresponds to rate of change. Great. But what happens when the line is curved? What happens if it's not a straight line, you see? What if it's a curve? How do we talk about change? How do we talk about slope? Well, slope is related to a line, which is tangent to the curve at a particular point. Now, suppose I have a point P here, and I want to talk about the slope of this tangent line. Well, how do I calculate that? I mean we know how to calculate slope if we have the coordinates of a couple of points. Okay. So let's talk about that. Let's create another point on this curve. Certainly if we have a function here, we have this graph, we can find the coordinates of that point. We can trace to that point for example on our graphing calculator and find the coordinates of the point. Let's call the point Q. And so we could say well, if we have a point out here that's Q and we have coordinates of that point and we have coordinates of this point. And certainly, by using our slope formula we can find the slope of the tangent line. We can even find the equation for the tangent line if we want to. Okay, great. Now, but does this tell us something about that? Well, not exactly, but it's an approximation. And if we make Q come a little bit closer, you see. If we put Q in this position and we have a closer approximation, well, what if we come even closer? We have a closer approximation, you see, by using closer and closer points. Now, how close can we get? That's where the limit idea comes in. We want to make this calculation or consider the idea as Q becomes infinitely close to P in this situation. And if we talk about approaching P with Q. Okay. Q and approach P, it is the limit as Q becomes infinitely close to P that will give us this tangent line. The slope of that tangent line at that particular point. So anyway, we'll revisit the problem a little bit later and we'll learn a great deal more about it. But incidentally this line, if we're talking about a couple of points, is called a secant line, and we're just making that secant line, you see, be a better and better approximator of the actual slope of the tangent line at that point. And it turns out by using limit we can get an exact value.