# 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?