The Epsilon-Delta Definition of a Limit

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”:

1. if “y tends to L” as “x tends to c.”
2. if “y approaches L” as “x approaches c.”
3. 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: