Limit IntuitionsDate: 10/01/98 at 02:04:51 From: Bryan J. Subject: Derivatives; definition of limit Dear Dr. Math, I have two questions: 1. In AP Calc AB, we just learned the formal definition of a function limit: lim f(x) = L such that |x-a| < alpha and |f(x)-L| < epsilon x-->a If I have explained it clearly from what I remember, could you please explain what that really means, because I have a little difficulty in understanding it. 2. I know enough about calculus to know that first derivative is slope. But I have wondered how to derive an expression of the equation of the tangent line of a function at a given point, in terms of the function and its derivatives. Could you help? Thanks. Date: 10/01/98 at 04:47:10 From: Doctor Mitteldorf Subject: Re: Derivatives; definition of limit Dear Bryan, 1) You have an intuitive idea of what a limit is. It's an interesting mental exercise to get that idea to agree with the formal definition. To start with, you haven't quoted the formal definition quite right. It is: For every epsilon there exists an alpha such that whenever x is within alpha of a, it's true that f(x) is also within epsilon of L. Another way to state this is: For every epsilon, there exists a alpha such that if |x-a| < alpha, then |f(x) - L| < epsilon. The intuitive idea you have is this: For some reason you can't calculate f(a). Maybe it comes out to be 0/0 or some other indeterminate form. But you can calcuate f(x) for x getting closer and closer to a. And as you do so, the values get closer and closer to some number L. Then you can say that L is the limit of f as x approaches a. What does this have to do with epsilon and alpha? Imagine you're trying to demonstrate to me that L is a good limit for f(x) as x gets close to a. I'm the skeptic. I tell you: I need f(x) to get within .001 of L, or else I'm not convinced. You do a little computation, and come back and say, "No problem. As long as x is close enough to a, you're guaranteed to be that close. In fact, by my computation if x is within .000375 of a, then f(x) is always within .001 of L." This computation is your proof. It's my responsibility to define "close", but then it's your responsibility to show that you can get that close to L, provided that x is sufficiently close to a. This takes a little concentration to follow, but it's actually the simplest formalization anyone's been able to come up with of our intuitive idea of what "limit" means. 2) This is quite straightforward. If you have a function f(x) at a point x, then the point on the plane is (x,f(x)) and the slope at that point is f'(x). The tangent line is of the form y = ax + b, where you know that the slope a = f'(x). What is b, then? It must be y - ax for that point, or b = f(x) - xf'(x). - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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