3.3 Unilateral limits and aysmptotes 14327 For hundreds of years, people have been interested in the magnitude of
7r(x), the number of primes less than or equal to x, when x is large. About the
year 1900, mathematicians succeeded in proving a remarkable fact that had been
surmised since the time of Euler (1707-1783). It was proved that
(*) lim a(x) = 1.
x w x
log x
We may know very little about logarithms and may not yet have learned that,
in mathematics above the level of elementary trigonometry, loglo x denotes the
logarithm of x with base 10 and log x denotes the logarithm of x with base e.
We may not yet know how to calculate log x when x is a given positive number.
Nevertheless we should be able to tell the meaning of the star formula. Do it.
Remark: Anyone who wishes to make a very modest calculation may use the fact
that log 20 is approximately 3 and may determine ir(20). When working on chalk
boards and scratch pads, many people make effective use of stars and daggers and
other things (instead of numbers) to designate significant formulas. The valuable
idea is illustrated only occasionally in this book.
28 It is sometimes said that mathematics is a language. Perhaps it would be
more sensible to say that mathematics is a collection of ideas and that mathe-
matics books use language in more or less successful attempts to reveal the ideas.
In any case, language is important and definitions constitute a basic part of this
language. To help us realize this fact, we consider an example involving regular
polygons. A regular polygon is a set in E2 consisting of the points on the line
segments PoPI, P1P2, ... , where the points P0, PI, , P, ,, Po are
equally spaced on a circle, n being an integer for which n > 3. Under this
definition, a circle is not a regular polygon. We do not have pencils sharp enough
to draw regular polygons having a million sides, but we can nevertheless tolerate
the idea that if we could draw one on an ordinary sheet of paper, then the result
would look like a circle. We cannot, however, tolerate the ancient collection of
words "a circle is a regular polygon having an infinite number of infinitesimally
small sides" as a part of our doctrine of limits. To take a sensible view of this
matter, we can know that there was a time when the best of our scientific ancestors
used fuzzy language and whale-oil lamps but we can also know that they worked
mightily to produce better products.
29 As was stated in Section 1.1, a number x appearing in this book is a real
number unless an explicit statement to the contrary is made. This circum-
stance does not prohibit recognition of the fact that numbers other than real
numbers can appear in mathematics. It is possible, and is sometimes worth-
while, to define and employ a set S* of numbers which contains each real number
x in the set S of real numbers and, in addition, two numbers - oo and om. When
the set S* is employed, each real number x is said to be finite and the numbers- oo and oo are said to be infinite (not finite). Order relations are introduced
in such a way that - oo < oo and - oo < x < co whenever x is a real number.
While these order relations are simple and attractive, it turns out to be impossible
to formulate a useful collection and algebraic laws (or postulates) in such a way
that oo - oo and 0 co are numbers in S*. Persons starting with enthusiasm
for - oo and co usually lose most of their fascination when they learn that the