# Formal Epsilon-Delta Definition of a Limit

- Author:
- Dr. Jack L. Jackson II

## Formal Epsilon-Delta Definiton of a Limit of a Function as x Approaches a Constant

Definition:
Given a function f and real numbers c and L, we say that if and only if
for any > 0 there exists a > 0 such that
if c - < x < c + , x c then L - < < L + .
This applet is designed to give a graphical interpretation of this formal definition.
Enter a desired function into the input box for f(x).
Choose a value for c by typing into its input box.
Determine the appropriate value for the limit L and type it into its input box.
Choose a value for epsilon () by adjusting its slider or typing into its input box.
Choose an appropriate corresponding value for delta () by adjusting its slider or typing into its input box.
If you can find an appropriate value of delta no matter how small the value of epsilon then you have demonstrated that the limit value L is correct.

## Question 1

What are the coordinates of the center of the purple box?

## Question 2

What are the height and width of the purple box?

## Question 3

What is the significance of the purple box? For a particular value of epsilon, how do we adjust delta so that the inequalities in the definition of the limit are satisfied?

## Question 4

For a particular value of epsilon, how can we see the largest corresponding value of delta so that the inequalities in the definition are satisfied?

## Question 5

What do we need to do if a smaller value of epsilon is chosen?

## Question 6

How can we tell from this exploration that the function has the limit that we specified?

## Question 7

How do we modify the definition and illustration for a right limit?

## Question 7

How do we modify the definition and illustration for a Left Limit?