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Formal Definition of a Limit

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Date: 10/17/2001 at 13:31:45
From: Rebecca Klusman
Subject: Calculus - the definiton of a limit

Could you please explain to me the formal definition of a limit. I
need help specifically with finding a delta for a given epsilon and
using the epsilon-delta definiton of a limit. The problems tell me
to find the limit and then find delta is greater than 0 such that
|f(x) - L| is less than 0.01 when ever 0 is less than |x - c| is less
than delta. The other set of problems tells me to find the limit L and
then use the epsilon-delta definition to prove that the limit is L.

I have no idea how to do this at all - it's quite over my head. I am
also taking this course by correspondence, so I have no one to help
me. Thank you!
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Date: 10/21/2001 at 15:27:22
From: Doctor Fenton
Subject: Re: Calculus - the definiton of a limit

Hi Rebecca,

Thanks for writing to Dr. Math. This isn't an easy concept to explain,
but see if the following helps.

The definition of a limit is a very sophisticated idea, and it's
important to be sure you understand what you're trying to do.

lim f(x) = L  means that by taking a value of x very x->a close, but
not equal, to a, the function value f(x) will be very close to L.
We need an explicit test for this idea.

First of all, how do you TEST the statement "x is close to a"? Whether
two numbers, say 5 and 6, are "close" is a matter of opinion, but
given another number, say 5.5, we can definitely say that 5.5 is
closer to 5 than 6 is. To decide if numbers are "close" we need a
"standard of closeness," which can be any positive number. We will
say that two numbers p and q are close according to the standard c if
|p-q|<c. For brevity, I will just say that p and q are c-close. So, if
c = .01, 5.2 and 5 are NOT .01-close, but 5.005 and 5 are .01-close,
because |5.2-5| = .2 > .01, while |5.005-5| = .005<.01 .

Think of the function f(x) as a "machine": a scientific or graphing
calculator is a good example. You push a key indicating the function
f, such as "sin" or "log," and type in a value of x, push "enter," and
the display changes to the value f(x) of the function f at x. Imagine
some machine generating random values of x that can be put into the
machine r. You are the "input quality control inspector," you must
decide whether to let the machine evaluate f(x), and you must base
your decision solely on whether x passes an "input closeness" test,
that is, whether x is close to a.

At the output end, an "output quality control inspector" will be
judging your work. She will have her own standard of closeness, and
she will test the output for closeness to L. If it passes, you get
paid; if it fails, you will be fired.

So, you will have a positive number for an input closeness standard
(the traditional name for the input closeness standard is the Greek
letter delta, which I will write d), and the output inspector will
have her own standard of closeness, traditionally called epsilon,
which I will write e.

There is one final consideration. Consider two functions f1 and f2,
where

f1(x)=x+1
and
f2(x)=2x+1.

Clearly, if x is very close to 2, then f1(x) is very close to 3, while
f2(x) is very close to 5, but we can be more precise. We can write

|f1(x)-3| = |x+1-3|

=|x-2| .

This equation says that the output closeness which the output
inspector will measure is exactly equal to the input closeness, for
the function f1.

For the function f2, we have

|f2(x)-7|=|2x+1-5|

= |2x-4|

= 2 |x-2|

which says that the "output closeness" is twice the input closeness.

In both cases, it is clear that there must be a relation between the
input and output closeness standards. For f1, if the output inspector
has a standard of closeness smaller than your input standard, some
inputs will pass your test but fail the output test, and you will be
fired. In fact, if the output inspector is out to get you fired, and
if she knows your input standard, she can come in after you have left,
run numbers that pass your input test through the machine until an
f(x) comes out that is not exactly equal to L, and then choose her
output standard so small that the output test will fail for that x.
This test will only be fair if the output inspector publicly announces
her output closeness standard, epsilon, before you have to choose your
input standard, delta.

So, if you are the inspector for function f1, and the output inspector
announces that epsilon = .01 will determine which outputs pass, you
know you may simply take your input closeness standard, delta, also
equal to .01, since you know, by your analysis of the machine, that
the output closeness is equal to the input closeness, so if x passes
the input test with delta = .01, the f1(x) will pass the output test
|f1(x)-3|<.01 . Now, if the output inspector retires, and a new output
inspector arrives determined to improve the quality of output, so that
from now on, epsilon will be .001, then you know you can keep your job
if you pick delta = .001. If you then get promoted to becoming input
inspector for f2, where the output inspector has a closeness standard
of epsilon = .0001, then your analysis of the machine f2 shows that
your input had better pass |x-2|<.00005, so you take delta = .00005.

The key is that no matter what output closeness standard, epsilon,
is chosen, it is always POSSIBLE to find an input closeness standard,
delta, that will guarantee that if the input x is delta-close to a,|x-
a|<d , then the output f(x) will be epsilon-close to L, |f(x)-L|<e.

That is, given any positive e, there exists a d such that any x
(different from a) which satisfies |x-a|<d will produce a value f(x)
that satisfies |f(x)-L|<e .

In a limit proof from the definition, your objective is to find a
strategy for choosing d after e is given. For most of the problems
usually encountered, you can find a relation between the input
closeness and the output closeness of the form

|f(x)-L|<= C|x-a| .

If you can find such an inequality, then it is easy to choose d when e
is given. For a function satisfying such an inequality, the output
closeness is no more than C times the input closeness, so you can
guarantee that the output closeness

|f(x)-L| < e

if the quantity C|x-a|<e, since C|x-a| is known to be larger than
|f(x)-L|.

Since C|x-a|<e if |x-a|<e/C, this suggests that we choose d as e/C.
Then, if |x-a|<d = e/C, then

|f(x)-L| <= C|x-a| < C[e/C] = e ,

and f(x) passes the output closeness test.

How do we find such an inequality for a given function? The functions
f1 and f2 above show that it is easy (actually an equality) for linear
functions f(x) = mx+b. What about f(x) = x^2 near a = 2?  We would
expect that
lim x^2 = 4
x->2             .

Look at the output closeness:

|f(x)-L| = |x^2 - 4|
= |x+2||x-2| .

Here, the output closeness is a variable multiple of the input
closeness. However, we are only interested in x's that are close to 2,
so we can make a preliminary restriction to consider only those x's
that are at least within 1 unit of 2. That is, we first require

|x-2|<1 .

If |x-2|<1, then -1 < x-2 < 1, or 1<x<3, and

3 < x+2 < 5,

so that the factor

|x+2| < 5.

Then, for x such that |x-2|<1,

|x^2-4| < 5|x-2|.

Given e>0, choose d so that both

|x-2| < 1  (to make the inequality true)

and so that

|x-2|<e/5, to make |f(x)-L| < e.

I hope this is of some help to you. If you have further questions

- Doctor Fenton, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Calculus

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