# Epsilon Delta Game (Limit of a function)

Topic:
Calculus
Play this game with either imaginary or real opponent. The game: Player 1: ﻿ ﻿ ﻿(1) Slides the point "MovePointA" along the x-axis to fix the postion of A and ﻿ ﻿ ﻿(2) Sets the value for . Player 2: ﻿Tries to set the value of so that the red portion of f(x) (i.e. all the values of f(x) ﻿in the interval ﻿(x-, x+) ) is completely in the green band. Player 1 wins if the player 2 is unable to do it. Otherwise, player 2 is the winner. Play with the applet and answer the questions below.
Questions: (don't forget to provide explanation!)
1. Which one would you want to be, player 1 or player 2?
2. If you are the player 1, where would you place the point A and what would be the value of to win the round? List a few winning combinations Point position & value.
3. Using green arrows in the upper right corner, scroll through the "challenge points" and decide if they could be winning or losing for player 1.
4. If you are the player 2 and you are allowed to fix the position of A at the beginning, where would you place it? (Your opponet would only be choosing the value of .)
Now think about how your winning and losing the game relates to the limit of f(x) at the point A. Challenge 9 and 10:
• What if A is "placed" at infinity and what if it is "placed" at negative infinity? Think about how the rules of the game should be altered in such case. Would you or your opponent be winning the game?
• What if the limit from the left and right are allowed? What should change in the applet and how would that affect the winning strategy?