3.3 Unilateral limits and asymptotes 135that is, f(x) exceeds M whenever a < x < a + 6. The assertion that
f(x) is large whenever x is large is abbreviated to
(3.352) lim f(x) = Co.
It means that to each number Al there corresponds a number N such that
(3.353) f(x) > M (x > N).
It is quite appropriate to recognize that ideas akin to those of this sec-
tion sometimes appear in elementary geometry books when information
about lengths of circles is being sought. Let C be a circle having radius #s
and diameter 1. We can imagine that, for each integer n >_ 3, we have
inscribed a regular polygon P. with n sides and have found its length L.
We can assume (or perhaps prove) that there is a number, which we can
call a, such that Ln is near 7r whenever n is large. By this we mean that
to each e > 0 there corresponds an integer N such that JL - 7rj < e
whenever n > N. The abbreviated form of the assertion is
(3.354) lira L = Tr.
n-ao
It is not necessary to try to explain how a polygon (which is something but
cannot do anything) can sprout more sides and approach the circle as the
number of "its" sides becomes infinite. The number it appearing in this
way is the length of circle of diameter 1. We are all familiar with the fact
that the length of a circle having radius r and diameter d is 27rr, or ud.
The ideas of this section have swarms of applications. In particular.
we can use them to introduce some ideas and terminology of analytic
geometry. We begin by considering the graph of the equation y = f (x),
where f is a given function. If
(3.36) lim f(x) = L or lim f(x) = L,
x-+m z-. - mthen the line having the equation y = L is called a horizontal asymptote
of the graph. If(3.361) lira f(x) = ao or lim f(x) _ -
or lira f(x) = co or lim f(x) = - oo,
x-+a- x-.a-then the line having the equation x = a is called a vertical asymptote of
the graph. Employing a modification of these ideas, we consider a case
in which -1 and B are numbers such that(3.37) lira [f(x) - (Ax + B)] = 0 or
x Olim U (x)(x) - (Ax + B)l = 0.
x--a