17 December 2004
How do we use limits?
by George E. Hrabovsky, President, MAST
Where We Have Been
Last time I wrote about infinity and integration. I showed what integration is and introduced the idea of a limit.
Last time we introduced the idea of a limit of a function. The notion of a limit is unique to calculus (or analysis, as it is called in its more advanced forms.) How, then, do we actually use it? According to [1] there seem to be two ways to use limits:
1) We can take the limit of a sequence of numbers.
2) We can take the limit of a function as the independent variable gets closer to a particular value.
This leads to three questions:
48. What is a sequence?
49. How do we take the limit of a sequence?
50. How do we take the limit of a function as the independent variable gets closer to a particular value?
What is a sequence?
According to [2] a sequence is an ordered set of mathematical objects. Often these objects are numbers and are denoted by curly braces. For example, the sequence of natural numbers can be written
There are many different kinds of sequences.
51. What different kinds of sequences are there?
What different kinds of sequences are there?
I will now introduce a nice little book by Donald Hight called A Concept of Limits [3]. There are two ways of classifying sequences. The first is to name the sequence according to its results. In this way you get the constant sequence, the arithmetic sequence, the geometric sequence, the Fibonacci sequence, and so on. The second way is to study the boundedness and monotonicity of the sequence.
52. What is a constant sequence?
53. What is an arithmetic sequence?
54. What is a geometric sequence?
55. What is the Fibonacci sequence?
56. What is the boundedness of a sequence?
57. What is the monotonicity of a sequence?
Before we get to the answer to this question we need to settle on some notation. We will define a sequence as a set of ordered pairs,
where  is the number of the element of the sequence, and  is the corresponding element of the sequence. The constant sequence can then be expressed by,
where  is some constant. In other words a sequence composed of n constants is a constant sequence.
If we have a sequence of numbers,
and we can express  as,
where  is the first term of the sequence and  is some given number. We can express this sequence as,
This is called an arithmetic sequence.
If we have a sequence of numbers,
and we can express  as,
where  is the first term of the sequence and  is some given number. We can express this sequence as,
This is called a geometric sequence.
If we have a sequence of numbers,
and we can express  as,
where the first two terms of the sequence is 1. We can express this sequence as,
This is called the Fibonacci sequence. We will discuss boundedness and monotonicity in the next issue.
This is a truly underrated book. It is a fantastic introduction to the central issue of calculus, the limit. It begins with a thorough and readable discussion of sequences and their limits. This is expanded to functions, and then considered in a general light. If you are confused about sequences, functions, or limits, this is the book for you.
[1] Richard A. Silverman, Modern Calculus and Analytic Geometry Macmillan Company, 1969 (republished in 2002 by Dover Publications).
[2] Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 1999.
[3] Donald W. Hight, A Concept of Limits, Prentice-Hall, Inc., 1966 (republished in 1977 by Dover Publications).
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(December 9, 2004)  |
