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### Limit Intuitions

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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
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.
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/
```
Associated Topics:
High School Analysis
High School Calculus

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