I am teaching honors calculus again in the Fall and I am very happy about that. Even though the first class is many months away, I am already thinking about how I am going to present various aspects of the material. In my experience, no matter how clearly one thinks he is presenting proofs, it takes a while for students to absorb the techniques and use them in a proper context. Here is another attempt to explain the basics of this subject matter clearly.
We say that if given there exists such that whenever . But what does this actually mean? When we say that , we are using mathematical notation to express the idea that gets closer and closer to as gets closer and closer to .
How do we make this intuitive idea mathematically precise? Being able to get arbitrarily close to means that the difference between and can be made as small as wish. Let us give the quantity “as small as we wish” a name and call it . This gets us started and tells that we wish for to be within of . But this is presumably going on while is getting closer and closer to . What does that mean?
Before going further, let us consider an example. Let and . Then the statement takes the form . I am sure that we all believe that indeed is getting closer and closer to as gets closer and closer to .
The first thing we do is decide how close we want our function to be to limit . This “how close” is typically measured using the quantity called . So we want to be within of , which in our case equals to . For to be within of , we must have
, which is equivalent to the statement
We can put everything together in the following way. Let be given. Let . Then whenever .
Let’s try a more elaborate example. Let’s compute
A quick glance tells us that the answer should be because and there are no apparent pitfalls. We need to make this observation formal, so let’s go for it.
Here , and . We want to be within a given threshold of . So we want
It bears repeating, with this observation in mind, that we want
, where is chosen depending on .
If in place of we had something like , we would know exactly what to do and set like before. But what do we do with the factor of ?
This means that if , then
If we insist that, among other constraints, , then
Let us now connect the dots. Let be given. Let
Since , , as we so above. Therefore
by definition of . This completes the proof.
How does one make sure that what is written above makes sense? Just work a couple of similar examples. For instance, prove that
Let us now consider a slightly more complicated example. We shall prove that
This is the same as proving that
and using the experience above as a guide we factor
This is done by computing the quotient . If you do not remember how to divide polynomials, please take a look at http://www.purplemath.com/modules/polydiv2.htm for a very nice and straightforward review of this important procedure.
Going back to our limit problem, given we must find such that whenever . Equivalently, we need
With the previous example as our guide, we look for an appropriate upper bound for keeping in mind that .
To do this, we write as a polynomial in . More precisely, we notice that
It follows that
If we insist, as before, that , then
since if .
We are now ready to put everything together. Let be given. Choose . Then
. This completes the proof.
Once again it is important to cement this idea and this can be done, for example, by proving that
A simple remark is in order here. Regardless of what polynomial we have on the left hand side, we know that if the limit is what we claim it is, then
and this means that we must be able to divide by . This is a consequence of a fact, which we are going to review in class, that if is a polynomial and , then
, where is a polynomial of one degree less.
Let us consider one more example where the main complication is to avoid accidentally dividing by . Consider
We all believe that the limit is equality , I am sure, and the trick is to carry out the argument in such a way that the fact that gets tricky as gets close to does not get in the way.
We want to make smaller than if . This is the same as showing that
With the previous examples as our guides, we know that the game is to argue that we may reduce to the case where we get to bound by some constant . We will then be able to set
. But what is in this case and how do we go about running the procedure? The enemy is the situation when is close to . The point is that if is small enough, then cannot possibly be very close to . For example, we may insist that . This implies that and we should be fine since is now guaranteed to be and we can take .
Let us now put everything together. Let be given. Set
. It follows that
and the proof is complete.
In order to make sure that this calculation is clear, prove that