The Epsilon-Delta Limit Game
- Author:
- L. Marizza A. Bailey, mrknauft
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?