Proof of Various Limit Properties
In this section we are going to prove some of the basic
properties and facts about limits that we saw in the
chapter. Before proceeding with any of the proofs we should note that many of
the proofs use the
definition of the limit and it is assumed that not only have you read that
section but that you have a fairly good feel for doing that kind of proof. If
you�re not very comfortable using the definition of the limit to prove limits
you�ll find many of the proofs in this section difficult to follow.
The proofs that we�ll be doing here will not be quite as
detailed as those in the precise definition of the limit section. The �proofs�
that we did in that section first did some work to get a guess for the
and
then we verified the guess. The reality is that often the work to get the guess
is not shown and the guess for
is
just written down and then verified. For the proofs in this section where a
is
actually chosen we�ll do it that way. To make matters worse, in some of the
proofs in this section work very differently from those that were in the limit
definition section.
So, with that out of the way, let�s get to the proofs.
Limit Properties
In the
Limit Properties section we gave several properties of limits. We�ll prove
most of them here. First, let�s recall the properties here so we have them in
front of us. We�ll also be making a small change to the notation to make the
proofs go a little easier. Here are the properties for reference purposes.
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Assume that
and
exist
and that c is any constant. Then,
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Note that we added values (K, L, etc.)
to each of the limits to make the proofs much easier. In these proofs we�ll be
using the fact that we know
and
we�ll
use the definition of the limit to make a statement about
and
which
will then be used to prove what we actually want to prove. When you see these
statements do not worry too much about why we chose them as we did. The reason
will become apparent once the proof is done.
Also, we�re not going to be doing the proofs in the order
they are written above. Some of the proofs will be easier if we�ve got some of
the others proved first.
Proof
This is a very simple proof. To make the notation a little
clearer let�s define the function
then
what we�re being asked to prove is that
.
So let�s do that.
Let
and
we need to show that we can find a
so
that
The left inequality is trivially satisfied for any x
however because we defined
.
So simply choose
to
be any number you want (you generally can�t do this with these proofs). Then,
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and
then we verified the guess. The reality is that often the work to get the guess
is not shown and the guess for
is
just written down and then verified. For the proofs in this section where a
is
actually chosen we�ll do it that way. To make matters worse, in some of the
proofs in this section work very differently from those that were in the limit
definition section.
and
exist
and that c is any constant. Then,
and
we�ll
use the definition of the limit to make a statement about
and
which
will then be used to prove what we actually want to prove. When you see these
statements do not worry too much about why we chose them as we did. The reason
will become apparent once the proof is done.
then
what we�re being asked to prove is that
.
So let�s do that.
and
we need to show that we can find a
so
that 
.
So simply choose
to
be any number you want (you generally can�t do this with these proofs). Then,
