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Limit Intuitions

Date: 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 

   lim f(x) = L  such that |x-a| < alpha and |f(x)-L| < epsilon

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? 


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   
Associated Topics:
High School Analysis
High School Calculus

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