he 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
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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
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
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 Derivative, Rate Of Change, Slope, Limits, Limit, Limitsection, Lthat, 1…

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