Page 70 - Calculus Demystified
P. 70
CHAPTER 2
Foundations of
Calculus
2.1 Limits
The single most important idea in calculus is the idea of limit. More than 2000 years
ago, the ancient Greeks wrestled with the limit concept, and they did not succeed.It
is only in the past 200 years that we have finally come up with a firm understanding
of limits. Here we give a brief sketch of the essential parts of the limit notion.
Suppose that f is a function whose domain contains two neighboring intervals:
f : (a, c) ∪ (c, b) → R. We wish to consider the behavior of f as the variable x
approaches c.If f(x) approaches a particular finite value as x approaches c, then
we say that the function f has the limit as x approaches c. We write
lim f(x) = .
x→c
The rigorous mathematical definition of limit is this:
Definition 2.1 Let a< c < b and let f be a function whose domain contains
(a, c) ∪ (c, b). We say that f has limit at c, and we write lim x→c f(x) = when
this condition holds: For each (> 0 there is a δ> 0 such that
|f(x) − | <(
whenever 0 < |x − c| <δ.
It is important to know that there is a rigorous definition of the limit concept, and
any development of mathematical theory relies in an essential way on this rigorous
definition. However, in the present book we may make good use of an intuitive
57
Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
