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09 May 2003

How Close Can We Get?

by George E. Hrabovsky, President, MAST

A Problem Involving a Rational Function

Suppose that you have analyzed data and determined that you have a function relating the data,

This is a reasonable function to understand. No real surprises until you decide to plot the graph of the function to see how it behaves. You discover something interesting. For and for the function behaves quite well. On the other hand, when the denominator becomes 0. This is a disastrous result! This means that the function has no meaningful value at the point corresponding to . What can we do?

Well, let's see what happens when we take values close to, but less than 3,

This seems to be getting closer and closer to the value 1.5. Now let's look at it from the other side of 3,

This also seems to be getting closer and closer to the value 1.5. We can't be completely sure of this, though, since all we have done is calculate some specific points near . What happens if we factor the numerator?

Now, factor the denominator

We see that there is a common factor of . Canceling this gives us,

When . We say that the limit of as approaches 3 is 3/2 (or, as we wrote above, 1.5). We write this,

When we have a function, the set from the smallest value of the range to the largest value of the range is called an interval. More formally, if we have the function, , defined from the values of to , then the interval of is from to . These values are called the end points of the interval.

There are three kinds of intervals. There are open intervals that do not contain their end points,

Another way to denote an open interval is the shorthand . There are also closed intervals that contain their end points,

These are also denoted . Then there are half-open intervals, where one end point is contained in the interval,

and

These can be denoted, and , respectively.

To determine if a function has a limit we need to ask a couple of questions:

1. As the independent variable, say x, gets closer to a particular value, say m where , does get closer to some real number, say L?
2. Does get as close as we want to L as x gets arbitrarily close to m so long as ?

If either of these answers are correct we can write,

or, more conventionally,

We say that the limit of approaches L as x approaches m.

The challenge was, how does the binomial coefficient help us with expressions of the form

This is solved through the use of the Binomial Theorem

If we look at the coefficients

and we look at the numerators

and we think about this long enough we will realize that these form an n-order factorial.

If we look at the denominators,

we see that each of these is the factorial of the term number,

we will call the term number . If we recall the definition of the binomial coefficient from last time,

This gives us the proper coefficients for the binomial theorem,

If we use the summation symbol and sum from to , we have

Can you define the trigonometric ratios for a triangle?

Online:

http://www.ping.be/~ping1339/lim.htm

This is not a graphically interesting page, but it seems to be very good for the math content.

or, for a more advanced treatment,

http://www.gap-system.org/~john/analysis/Lectures/L13.html

Books:

Richard A. Silverman, 1969, Modern Calculus and Analytic Geometry, MacMillan Company, New York (Dover Publications has reprinted this book with corrections in 2002). This has a very nice chapter on limits.

Converted by Mathematica  (May 9, 2003)