Untitled Document

23 May 2003

Those Little Details

by George E. Hrabovsky, President, MAST

•Another Look at Our Problem Involving a Rational Function

Recall from last time that you have analyzed data and determined that you have a function relating that data,

You discovered that for and for the function behaves quite well, but when the denominator becomes 0. We then went on to develop the notion of a limit.

Let's take a closer look at what happens when we take values close to 3. Let us begin with an interval of ,

Note that in each case the interval of y from 1.5 is . From long tradition we will adopt the Greek letter delta, , for the x interval and the Greek letter epsilon, , for the y interval. For the case above we say that for . We can then say that as the value of x gets to within of 3 the limit of the function gets to within of 1.5. If

so, and , so . If

then . Finally, if

then .
Another way to look at it is that if

then

Another way of looking at this, is if

then, for every there is a such that if is in the open interval and , then must be in the open interval .

•Absolute Values and Inequalities

From basic algebra recall that an absolute value is defined as

We can relate the following properties of absolute values;

$FormBox[RowBox[{| x |^2, , =, , RowBox[{x^2, Cell[ ... bsp; ], (4)}]}], TraditionalForm]$

$FormBox[RowBox[{| x _ 1 - x _ 2 |, , =, , RowBox[{Overscript[P _ 1 P _ 2, _], , =, , RowBo ... ; ], (12)}]}]}], TraditionalForm]$

$FormBox[RowBox[{| x _ 1 |, , =, , RowBox[{distance, , from, , the, , origin, , to, ... sp; ], (13)}]}], TraditionalForm]$

where (8) and (9) define the relationships between absolute values and intervals. (11) is the famous triangle inequality.

•The Formal Definition of a Limit

Using the notation, we can make our definition of the limit from last time more precise. Assume we have some function that is defined on some open interval such that . Then, if we have the real numbers and such that

then we have

where is called the limit point. We can see the first condition in the diagram below,

[Graphics:Images/index_57.gif]

the second condition is described in the diagram below,

[Graphics:Images/index_58.gif]

together these give us the situation in the diagram below,

[Graphics:Images/index_59.gif]

The regions in blue are sometimes called neighborhoods around the limit point or the limit. Next time we will discuss how we can have limits that change depending on what direction you approach the limit point from.

•The Math Challenge from Last Time

The challenge was to define the trigonometric ratios. Many people start with a triangle to show these relationships. I prefer a unit circle (that is a circle of radius 1 in whatever units of distance are being used). In the diagram below I have drawn a circle, and I have introduced a standard coordinate system at the origin. I have also chosen a point, , on the circle and have drawn a line segment, , from the origin to . The angle formed by the intersection of and shall be denoted by the greek letter theta, .

[Graphics:Images/index_66.gif]

We can now define our trigonometric ratios (note that all and values are for the location of ). We begin with the sine function,

The cosine function is

The tangent function is,

The cotangent function is then,

The secant function is,

Finally, the cosecant function is,

Since we are using a unit circle, these ratios become,

•The Math Challenge

Can you prove the triangle inequality and interpret it? Can you show how the trigonometric ratios are related to each other?

•Math Resources for Limits

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 23, 2003)