This chapter begins with a brief historical introduction, wherein the seeming
paradox of Achilles and the turtle is examined. Although this treatise does follow part of the
tradition and progresses to the concepts of limits and continuity, including the more formal
perspective of using epsilon and delta type proofs that have been the touchstone of calculus’s
foundation for over three centuries, no further development of such a protocol is undertaken.
In its place a very different “Weltanschauung” (world philosophy) that focuses on what this
author asserts is the appropriate underlying foundation of calculus is promulgated; namely,
the relation of the concepts of none, some and all (algebraically expressed as 0, 1 and ∞) to
the six fundamental operations of numbers (addition, subtraction, multiplication, division,
raising to a power and extracting a root). From the 54 potential binary combinations of these
sets, the seven traditional indeterminate l’Hôpital forms, as well as three additional related
forms that mathematicians have missed for over three centuries are distilled. In the process,
attention is focused on combinations deliberately disallowed in previous mathematics
courses; especially those that arise with respect to infinity and division by zero. One
particular combination, which has as its objective the determination of those extreme values
that the given function can reach both globally (over all of space), and locally (in a given
interval), is postulated to be the foundation upon which, provided the appropriate constraints
are included, the first of the major techniques of calculus is to be built. The philosophy
espoused herein views a specific related function, derived from the given function and thus
named as “the derivative of that function”, as the division of two, considered to be even more
elementary, functions, called “differentials”. Each of these differentials, which are primarily
algebraic constructs, is equivalent to having a limit value of 0. Consequently, the derivative
may be viewed as giving meaning to the indeterminate form
0\0 , under a set of constraints to be designated at a later time. Meanwhile, selected other entities, which had been historically
defined, such as the concept traditionally expressed as “concavity”, are viewed as having
been relegated to the status of insignificance. This is, in contradistinction to many traditional calculus textbooks which belabor concavity as being nearly equal in importance with the
extreme values of maxima and minima. The topological subtleties, often forming the basis of
theoretically biased courses, are included only when they add to an intuitive understanding of
the subject matter, and thus become of interest to applied scientists and engineers.
Two other l’Hôpitalian combinations, which are similarly depicted as forming the foundation
for the other two significant terms that comprise the principal domain associated with
calculus will be introduced and developed in Chapters 4 and 6 respectively.
Keywords: Arithmetic Operations Involving Infinity, Continuity, Curve Sketching,
Derivatives (Definition, Poly- vs. Multi-nomials, Trig Functions), Differential
Calculus, Epsilon-Delta Processes, Extrema (Maximum, Minimum, Point of
Inflection), Implicit Differentiation, “Last” Number and Interpreting Infinity:
l’Hôpital Indeterminate Forms vs. l’Hôpital-Elk, Indeterminate Forms, Limits,
Related Rates.
