he Definition of the Limit
In this section we’re going to be taking a look at the precise, mathematical definition
of the three kinds of limits we looked at in this chapter. We’ll be looking at the
precise definition of limits at finite points that have finite values, limits that are
infinity and limits at infinity. We’ll also give the precise, mathematical definition of
continuity.
Let’s start this section out with the definition of a limit at a finite point that has a finite
value.
Definition 1
Let
f(x)
be a function defined on an interval that contains
, except
possibly at
. Then we say that,
if for every number
there is some number
such that
Wow. That’s a mouth full. Now that it’s written down, just what does this mean?
Let’s take a look at the following graph and let’s also assume that the limit does exist.
What the definition is telling us is that for
any
number
that we pick
we can go to our graph and sketch two horizontal lines at
and
as shown on the graph above. Then somewhere out there in the world is
another number
, which we will need to determine, that will allow us
to add in two vertical lines to our graph at
and
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentNow, if we take any
x
in the pink region,
i.e.
between
and
, then this
x
will be closer to
a
than either of
and
.
Or,
If we now identify the point on the graph that our choice of
x
gives then this point on
the graph
will
lie in the intersection of the pink and yellow region. This means that
this function value
f(x)
will be closer to
L
than either of
and
. Or,
So, if we take any value of
x
in the pink region then the graph for those values
of
x
will lie in the yellow region.
Notice that there are actually an infinite number of possible
δ
€
’s that we can choose.
In fact, if we go back and look at the graph above it looks like we could have taken a
slightly larger
δ
and still gotten the graph from that pink region to be completely
contained in the yellow region.
Also, notice that as the definition points out we only need to make sure that the
function is defined in some interval around
but we don’t really care if
it is defined at
. Remember that limits do not care what is happening
at the point, they only care what is happening around the point in question.
Okay, now that we’ve gotten the definition out of the way and made an attempt to
understand it let’s see how it’s actually used in practice.
These are a little tricky sometimes and it can take a lot of practice to get good at these
so don’t feel too bad if you don’t pick up on this stuff right away. We’re going to be
looking a couple of examples that work out fairly easily.
Example 1
Use the definition of the limit to prove the following limit.
Solution
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 sc
 Limits, Limit of a function, Negative and nonnegative numbers, Lthan

Click to edit the document details