The Epsilon-Delta Method

Introduction

To prove a limit exists, we prove that as

approaches , approaches .

There are many methods to prove this. The first is the epsilon/delta method. Each numbered paragraph is a different phrasing of that method. Being stated three times, in three different ways, will help it click.

1. For any range around the limit,

c

, we need to prove there exists at least one range around

a

, where all the values of

f(x)

is within the first range as long as all the values of

x

are in the second range.

In the graph below, the limit being proven is , which means that as approaches , approaches . The blue lines represent a range around the limit, . The red lines represent a range of -values around where, as long as is within the red lines, is within the blue lines.

2. For any set of blue lines, there is a set of red lines where, as long as the

x

-values are within the red lines, the

f(x)

-values are within the blue lines.

Symbols: Epsilon - . Delta - .

3.

\forall \epsilon \gt 0, \exists \delta \gt 0

where if

(a - \delta \lt x \lt a + \delta)

, then

(c - \epsilon \lt f(x) \lt c + \epsilon)

.

3 (again). For all epsilon greater than

0

, there exists delta greater than

0

, where if the distance between

x

and

a

is less than delta, then the distance between

f(x)

and

c

is less than epsilon. For example, for , I can prove that with , if , then . Here it is zoomed in. And zoomed even further in.

Cleaning the Notation

The two statements below are the same. The first one better matches the visualizations above, but we will be using the second from this point on.

where if , then .

where if , then .