After you have had an opportunity to do a few limit computations by plugging away with a calculator, you can begin to appreciate the limitations of the numerical method as a means for solving problems.
One of the primary goals of mathematics is to develop better, more general methods for solving problems. Often one arrives at such methods only after carrying out a process of abstraction. The purpose of abstraction is to focus our attention on the most important details in a problem or set of related problems while ignoring unimportant details of the problem.
Abstraction is a very important tool for doing mathematics, but has to be handled with some care. In particular, when we deal with very abstract problems we often lose the connection to hard, verifiable reality that more concrete problems provide us. This in turn can make it much more difficult to verify whether or not the answers we are coming up with are correct. This heightens the need for some sort of rigorous formal proof proceedure that will allow us to verify the correctness of our results.
A mathematical proof is essentially a series of statements in which each statement follows from the previous statements according to the rules of logic. Every proof system starts with a set of statements called axioms that can not be proved from other statements and have to be taken as given. Often sets of axioms are somewhat fluid. One author may count a particular statement as an axiom and use it to prove another statement, while another author may decide to declare the second an axiom and use it to prove the first.
In an effort to keep the set of axioms as small as possible, many authors start with a very small set of axioms and immediately use them to prove a number of other seemingly obvious statements as theorems. After establishing a few of these elementary results as theorems authors then go on to prove theorems that deal with ideas that are more substantial and general.
Last week I showed you some of the fundamental rules for working with exponentials and logarithms. It is possible to organize those rules into axioms and theorems and use the axioms to prove the theorems. Here is one possible organization.
Axioms:
Theorems:
You may be surprised to see a0 = 1 listed among the theorems rather than the axioms. Here is a proof of this theorem:
Here is a set of basic rules governing the behavior of limits that will take as given.
Theorem (The limit of a constant is a constant) If c is a constant and f(x) is a function with the characteristic that f(x) is defined and equal to c for all x not equal to a then the limit
exists and equals c.
Theorem The limit
exists for all a and equals a.
Theorem Let c be a constant and Let f(x) and g(x) be two functions for which the limits
both exist. Then
Why are these listed as theorems and not axioms? It turns out that all of these limit rules can be proved from a more fundamental definition. In section 2.4 the author proves all of these from the formal definition of the limit. That section is a little too advanced for this course, but if you are adventurous you can read that section on your own.
For convenience below, let's summarize the limit rules and number them for easy reference.
| (1) |
| (2) |
| (3) |
| (4) |
| (5) |
| (6) |
| (7) |
Let's see how these rules get applied in practice. Consider the problem
To compute this limit, we can try the algebraic trick of factoring and cancelling to make the 0/0 problem go away.
6
Now that we have established to our satisfaction that the limit is 6, let us try proving this formally as an exercise in writing proofs. We start with a statement of the theorem to prove.
Theorem
How do we write a proof? Basically, we repeat the steps shown above, being careful to provide the appropriate justification for each step. Further, the setup of the limit rules make it better to proceed in reverse, because most of the rules work on the assumption that the limits of two functions f(x) and g(x) exist at a. That requirement essentially forces us to work backwards through the steps.
To start with, we have
by applications of rules (2) and (1). Rule (3) gives
Next comes a clever application of rule (1). The function
is defined and equal to 1 at every point except x = 3. This follows the precise statement of rule (1) and allows us to say that
Thus
An application of rule (6) leads us to
Putting this all together now we have a chain of equalities that prove that
and the theorem is proved.
The algebraic methods we have developed are useful for solving a wide range of problems, but still will not help us to solve all problems. A good example is
To handle this problem we need one more theorem:
The Squeeze Theorem If 
for all x near a and
then
Because
for all x, it is easy to see that
for all x near 0. Since
the squeeze theorem says
Section 2.3: 1, 2, 4, 5, 10, 47, 48, 60
