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Graphing Limits


Date: 09/12/2001 at 13:43:47
From: Justin Morrisroe
Subject: Calculus-Limits

I was hoping you could explain the concept of "limits" and how to read 
them from graphs, and why substituting the "c"# into the equation 
doesn't always work 

(i.e. lim   (x^2-1)/(x+1) )
      x-->1 

the answer is 2, but I din't get 2, I got undefined. 

Help, please!


Date: 09/12/2001 at 15:01:27
From: Doctor Peterson
Subject: Re: Calculus-Limits

Hi, Justin.

Here's the basic idea of a limit: We have a function, which may or may 
not be defined at x = c. We want to know how it behaves near x = c, 
so we look at values of x closer and closer to c. Suppose the graph of 
y = f(x) looks like this:

    y
    |        o           :
    |               o    :
    |                  o :
   L+ - - - - - - - - - -?- - - - - - -
    |                    :  o
    |                    :        o
    |                    :
    +--------------------+-------------- x
                         c

The closer x gets to c, the closer y gets to L. Then we say that f(x) 
approaches L as x approaches c; that is, L is the limit of f(x) as x 
approaches c. For the sort of functions we usually work with, L is in 
fact the value of f(c), so this seems like a waste of effort. But in 
some very important cases (those for which this concept was invented), 
f(c) is not even defined, so the limit is the only way we can talk 
about this idea of closeness. In a sense, the limit "fills the gap" in 
the definition of the function, allowing us to assign a value to f(c) 
that makes the function "continuous."

The trick is to define "closer and closer" precisely. We do this by 
seeing it as sort of a game, the sort of game you might have between 
two gamblers who don't trust one another. You say it's close - but how 
do I know that what you mean by "close" is really close enough?

So I make a challenge: I'll believe that L is the limit, if for ANY 
definition of "close" I choose (that is, any distance I call "close 
enough"), you can give me a range around x = c within which the 
function is always that close to L. That's what the "epsilon-delta 
definition" of a limit means. If _you_ can keep the function as close 
to L as _I_ want it to be, I'll believe you. (It's sort of like having 
one kid cut the cake and the other choose the larger piece.)

Now, I'm a little confused about your example; you probably typed 
something wrong. If I replace x with 1, I get 0/2, which is just 0. A 
more interesting problem is

     lim (x^2-1)/(x-1)
    x-->1

To find this limit, you have to simplify the expression so that it is 
defined for x = 1, and then substitute. Without simplifying, you get 
0/0, which is indeterminate. (That is, it might have any value at all, 
and can't be determined without the additional information in the 
limit.) So this is an example where the function is not defined at 
x = 1, yet we can talk about its limit there. What we are doing is 
showing that the function is equal to a new function everywhere except 
at x = 1, so we can "fill the gap" by saying it is also equal to that 
function at x = 1. This value is the limit.

I think you may find this discussion useful:

   Understanding the Need for Limits
   http://mathforum.org/dr.math/problems/josh.2.19.01.html   

Here's another point to consider: the term "limit" is probably the 
source of much of the confusion over this concept, because it doesn't 
really fit this setting. It was originally used, I believe, of the 
limit of a sequence, where the terms a[i] approach some "limiting 
value" when i becomes infinite. (For example, the sequence a[i] = 1/i 
approaches zero as i becomes infinite.) The essential concept is the 
same, but there it makes more sense to call it a limit, because the 
terms are going closer to that value but never go beyond it. It's 
better at first to talk about "approaching L" rather than "having 
limit L" for the sake of clarity.

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

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