The Limit
In the previous
section we looked at a couple of problems and in both problems we had a
function (slope in the tangent problem case and average rate of change in the
rate of change problem) and we wanted to know how that function was behaving at
some point
 .
At this stage of the game we no longer care where the functions came from and we
no longer care if we�re going to see them down the road again or not. All that
we need to know or worry about is that we�ve got these functions and we want to
know something about them.
To answer the questions in the last section we choose
values of x that got closer and closer to
 and
we plugged these into the function. We also made sure that we looked at values
of x that were on both the left and the right of
 .
Once we did this we looked at our table of function values and saw what the
function values were approaching as x got closer and closer to
 and
used this to guess the value that we were after.
This process is called taking a limit and we have
some notation for this. The limit notation for the two problems from the last
section is,
In this notation we will note that we always give the
function that we�re working with and we also give the value of x (or t)
that we are moving in towards.
In this section we are going to take an intuitive approach
to limits and try to get a feel for what they are and what they can tell us
about a function. With that goal in mind we are not going to get into how we
actually compute limits yet. We will instead rely on what we did in the
previous section as well as another approach to guess the value of the limits.
Both of the approaches that we are going to use in this
section are designed to help us understand just what limits are. In general we
don�t typically use the methods in this section to compute limits and in many
cases can be very difficult to use to even estimate the value of a limit and/or
will give the wrong value on occasion. We will look at actually computing
limits in a couple of sections.
Let�s first start off with the following �definition� of a
limit.
Definition
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We say that the limit of f(x) is L as
x approaches a and write this as
provided we can make f(x) as close to L
as we want for all x sufficiently close to a, from both
sides, without actually letting x be a. |
This is not the exact, precise definition of a limit. If
you would like to see the more precise and mathematical definition of a limit
you should check out the
The
Definition of a Limit section at the end of this chapter. The definition
given above is more of a �working� definition. This definition helps us to get
an idea of just what limits are and what they can tell us about functions.
So just what does this definition mean? Well let�s suppose
that we know that the limit does in fact exist. According to our �working�
definition we can then decide how close to L that we�d like to make
f(x). For sake of argument let�s suppose that we want to make f(x)
no more that 0.001 away from L. This means that we want one of the
following
Now according to the �working� definition this means that
if we get x sufficiently close to we can make one of the above true.
However, it actually says a little more. It actually says that somewhere out
there in the world is a value of x, say X, so that for all x�s
that are closer to a than X then one of the above statements will
be true.
This is actually a fairly important idea. There are many
functions out there in the work that we can make as close to L for
specific values of x that are close to a, but there will other
values of x closer to a that give functions values that are
nowhere near close to L. In order for a limit to exist once we get
f(x) as close to L as we want for some x then it will need to
stay in that close to L (or get closer) for all values of x that
are closer to a.
In somewhat simpler terms the definition says that as x
gets closer and closer to x=a (from both sides of course�) then f(x)
must be getting closer and closer to L. Or, as we move in towards
x=a then f(x) must be moving in towards L.
It is important to note once again that we must look at
values of x that are on both sides of x=a. We should also note
that we are not allowed to use x=a in the definition. We will often use
the information that limits give us to get some information about what is going
on right at x=a, but the limit itself is not concerned with what is
actually going on at x=a. The limit is only concerned with what is going
on around the point x=a. This is an important concept about limits that
we need to keep in mind.
An alternative notation that we will occasionally use in
denoting limits is
How do we use this definition to help us estimate limits?
We do exactly what we did in the previous
section. We take x�s on both sides of x=a that move in closer
and closer to a and we plug these into our function. We then look to see
if we can determine what number the function values are moving in towards and
use this as our estimate.
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