## Lemma and Definition for Limits

LEMMA

(i) x is within δ of c if and only if

c - δ ≤ x ≤ c + δ.

(ii) x is strictly within δ of c if and only if

c - δ < x < c + δ.

PROOF (i) Subtracting c from each term we see that

c - δ ≤ x ≤ c + δ

if and only if

- δ ≤ x - c ≤ δ,

which is true if and only if

|x - c| ≤ δ.

The proof of (ii) is similar.

We shall repeat our infinitesimal definition of limit from Section 3.3 and then write down the ε, δ condition for limits. Later we shall prove that the two definitions of limit are equivalent to each other.

Suppose the real function f is defined for all real numbers x ≠ c in some neighborhood of c.

DEFINITION OF LIMIT (Repeated)

The equation

limx→c f(x) = L

means that whenever a hyperreal number x is infinitely close to but not equal to c, f(x) is infinitely close to L.

ε, δ CONDITION FOR limx→c f(x) = L

For every real number ε > 0 there is a real number δ > 0 which depends on ε such that whenever x is strictly within δ of c but not equal to c, f(x) is strictly within ε of L. In symbols, if 0 < |x - c| < δ, then |f(x) - L| < ε.

In the ε, δ condition, the notion of being infinitely close to c is replaced by being strictly within δ of c, and being infinitely close to L is replaced by being strictly within ε, of L. But why are there two numbers ε and δ, instead of just one? And why should δ depend on ε? Let us look at a simple example.

 Example 1
 Example 2

Last Update: 2006-11-05