In this section we are going to prove some of the basic
properties and facts about limits that we saw in the Limits
chapter. Before proceeding with any of
the proofs we should note that many of the proofs use the precise 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.
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 of 7
Proof of 1
Proof of 2
Note that we’ll need something called the triangle inequality in this proof. The triangle inequality states that,
Here’s the actual proof.
Proof of 3
This one is a little tricky. First, let’s note that because and we can use 2 and 7 to prove the
following two limits.
Now, let . Then there is a and a such that,
Choose . If we then get,
So, we’ve managed to prove that,
This apparently has nothing to do with what we actually
want to prove, but as you’ll see in a bit it is needed.
Before launching into the actual proof of 3 let’s do a little Algebra. First, expand the following product.
Rearranging this gives the following way to write the
product of the two functions.
With this we can now proceed with the proof of 3.
Fairly simple proof really, once you see all the steps
that you have to take before you even start.
The second step made multiple uses of property 2. In the third step we
used the limit we initially proved. In
the fourth step we used properties 1
and 7. Finally, we just did some simplification.

Proof of 4
This one is also a little tricky. First, we’ll start of by proving,
Let .
We’ll not need this right away, but these proofs always start off with
this statement. Now, because there is a such that,
Now, assuming that we have,
Rearranging this gives,
Now, there is also a such that,
Choose . If we have,
Now that we’ve proven the more general fact is easy.

Proof of 5. for n an integer
As noted we’re only going to prove 5 for integer exponents.
This will also involve proof by induction so if you aren’t familiar
with induction proofs you can skip this proof.
So, we’re going to prove,
For we have nothing more than a special case of
property 3.
So, 5 is proven
for . Now assume that 5 is true for ,
or
. Then, again using property 3 we have,

Proof of 6
As pointed out in the Limit
Properties section this is nothing more than a special case of the full
version of 5 and the proof is
given there and so is the proof is not give here.

Proof of 8
Proof of 9
This is just a special case of property 5 with and so we won’t prove it here.

Facts, Infinite Limits
Partial Proof of 1
Proof of 2
Proof of 3
Let then because we know there exists a such that if we have,
Note that because in the case we will have here.
Next, because we know there exists a such that if we have,
Again, because we know that we will have . Also, for reasons that will shortly be
apparent, multiply the final inequality by a minus sign to get,
Now, let and so if we know from the above statements that we
will have both,
This gives us,
This may seem to not be what we needed however multiplying
this by a minus sign gives,
and because we
originally chose we have now proven that .

Proof of 4
We’ll need to do this in three cases. Let’s start with the easiest case.
Case 1 :
Let then because we know there exists a such that if we have,
Next, because we know there exists a such that if we have,
Now, let and so if we know from the above statements that we
will have both,
This gives us,
In the second
step we could remove the absolute value bars from because we know it is positive.
So, we proved that if .
Case 2 :
Let then because we know there exists a such that if we have,
Next, because we know there exists a such that if we have,
Also, because we are assuming that it is safe to assume that for we have .
Now, let and so if we know from the above statements that we
will have both,
This gives us,
In the second step we could remove the absolute value bars
because we know or can safely assume (as noted above) that both functions
were positive.
So, we proved that if
.
Case 3 : .
Let then because we know there exists a such that if we have,
Next, because we know there exists a such that if we have,
Next, multiply this be a negative sign to get,
Also, because we are assuming that it is safe to assume that for we have .
Now, let and so if we know from the above statements that we
will have both,
This gives us,
In the second step we could remove the absolute value bars
by adding in the negative because we know that and can safely assume that (as noted above).
So, we proved that if
.

Fact 1, Limits At
Infinity, Part 1
1. If r is a positive rational number and c is any real number then,
2. If r is a positive rational number, c is any real number and x^{r} is defined for then,

Proof of 1
Proof of 2
Fact 2, Limits At
Infinity, Part I
Proof of
Proof of
Fact 2, Continuity
Proof