Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

MORE ABOUT LIMITS 283
F under either D4 or D3 but that, under D2, B and C as well
as D are limits of F.
With the foregoing non-quantitative-purely serialconception of limit, you can make yourselves familiar
by applying the definition to numerous examples which
you can readily construct or easily find, for they abound
on every hand.
I have now spoken of limits and limit conceptions at
far greater length than I had originally intended to do.
If I have thus exhausted your interest and patience, I
assure you that I have by no means exhausted the subject.
There are in use yet other conceptions of the term limit
and connected therewith many interesting and important
refinements,-refinements of refinements,-with which,
however, I do not intend to trouble you. There remain
two questions which must have occurred to you and
which I am sure you will desire to consider before we
take final leave of the subject. One of them is easy and
admits of a brief answer. The question is: In view of the
variety of senses in which mathematicians employ the
term " limit," how do they manage, if they do manage,
to avoid confusion-confusion of themselves and others?
The answer is: They do not always avoid it, but in general
they do, and they do so, as I have already intimated, by
indicating either explicitly or contextually, when speaking
of a limit, the sense in which the term is to be understood.
The second question relates to the scientific and philosophic importance of the term. Both by dwelling on it so
long and by explicit statement, I have said that its importance is very great. I wish now to show that the estimate
is just and how it is so.
Scientific and Philosophic Importance of the Term
Limit.-As to its scientific importance, the task of show