by courtesy of the author:
APEX Calculus I/II/III
University of North Dakota
Adapted from APEX Calculus
by Gregory Hartman, Ph.D., Department of Applied Mathematics, Virginia Military Institute
1.2 Epsilon-Delta Definition of a Limit
This section introduces the formal definition of a limit.
Many refer to this as “the epsilon-delta,” definition, referring to the letters ϵ and δ of the Greek alphabet.
Before we give the actual definition, let’s consider a few informal ways of describing a limit. Given a function y=f(x) and an x-value, c, we say that “the limit of the function f, as x approaches c, is a value L”:
- if “y tends to L” as “x tends to c.”
- if “y approaches L” as “x approaches c.”
- if “y is near L” whenever “x is near c.”
The problem with these definitions is that the words “tends,” “approach,” "near",
and especially “near” are not exact.
In what way does the variable x tend to, or approach, c?
**How near **do x and y have to be to c and L, respectively?
The definition we describe in this section comes from formalizing 3. A quick restatement gets us closer to what we want:
?′. If x is within a certain tolerance level of c,
then the corresponding value y=f(x) is within a certain tolerance level of L.
The traditional notation for the x-tolerance is the lowercase Greek letter delta, or δ, and the y-tolerance is denoted by lowercase epsilon, or ϵ. One more rephrasing of ?′ nearly gets us to the actual definition:
?′′. If x is within δ units of c, then the corresponding value of y is within ϵ units of L.
We can write “x is within δ units of c” mathematically as
|x-c|<δ, which is equivalent to c-δ<x<c+δ.
Letting the symbol “⟶” represent the word “implies,” we can rewrite ?′′ as
|x-c|<δ⟶|y-L|<ϵ or c-δ<x<c+δ⟶L-ϵ<y<L+ϵ.
The point is that δ and ϵ, being tolerances, can be any positive (but typically small) values.
Finally, we have the formal definition of the limit with the notation seen in the previous section.
Definition 1.2.1 The Limit of a Function
Let I be an open interval containing c , and let f be a function defined on I , except possibly at c . The limit of f(x), as x approaches c, is L , denoted by
lim x→c f(x) = L,
means that given any ϵ>0, there exists δ>0 such that for all x≠c, if |x-c|<δ, then |f(x)-L|<ϵ.