Pauls Online Notes : Calculus I - Proof of Various Limit Properties

Paul's Online Math Notes

Calculus I

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Extras	E-Book Chapter Section	Proof of Various Derivative Properties

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 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.

Assume that and exist and that c is any constant. Then,

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

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,

Proof of 1

There are several ways to prove this part. If you accept 3 And 7 then all you need to do is let and then this is a direct result of 3 and 7. However, we’d like to do a more rigorous mathematical proof. So here is that proof.

First, note that if then and so,

The limit evaluation is a special case of 7 (with ) which we just proved Therefore we know 1 is true for and so we can assume that for the remainder of this proof.

Let then because by the definition of the limit there is a such that,

Now choose and we need to show that

and we’ll be done. So, assume that and then,

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.

We’ll be doing this proof in two parts. First let’s prove .

Let then because and there is a and a such that,

Now choose . Then we need to show that

Assume that . We then have,

In the third step we used the fact that, by our choice of , we also have and and so we can use the initial statements in our proof.

Next, we need to prove . We could do a similar proof as we did above for the sum of two functions. However, we might as well take advantage of the fact that we’ve proven this for a sum and that we’ve also proven 1.