# The Epsilon-Delta Limit Game

## The Formal Definition of Limit

The limit of as is if and only if for every distance, , there is another distance, , such that if

*x*is within delta of*c*, , then*f*(*x*) is within epsilon of*f*(*c*), . This game will help you understand the statements above.## Example 1

The function below is
We claim that .
The epsilon is given. Try to find a suitable delta (small enough), so that the image of the set of all x's such that (which are represented by where the graph of

*f*intersects the green vertical band), are within the set of*f*(*x*)'s such that (which is represented by where the graph of*f*intersects the red horizontal band)## Which Delta won?

Write the value of delta that helped you win the limit game.

## Red and Green Bands

What was different with the red and green band when you won?

## Change Epsilon to something smaller.

State the new value of epsilon you chose. What delta worked for that epsilon? How did it compare to the previous delta?

## Example 2

The function below is
We claim that .
The epsilon is given. Try to find a suitable delta (small enough), so that the image of the set of all x's such that (which are represented by where the graph of

*f*intersects the green vertical band), are within the set of*f*(*x*)'s such that (which is represented by where the graph of*f*intersects the red horizontal band)## Did you win?

Were you able to find a suitable delta? Explain?

## Change the epsilon?

Did changing the epsilon help you find a suitable delta? Why or Why not?

## Change L to 29.

Can you find a suitable delta now? Explain.

## Red and Green Bands

What is going on with the red and green bands in this example? How do they illustrate that the limit doesn't exist?