The Epsilon Delta Game

I.  Quiz

II.  Homework

III.  The rules of the game

The students in the class will pair up and play the epsilon delta game as is stated in the book.  The instructor will play the game with three or four examples to show show some strategies.

IV.  The formal definition of the limit.

Let f(x) be a function and L be a number we say that

lim as x -> a f(x) = L

if for any choice of epsilon, the delta team can respond with a positive number delta so that with a "perfect calculator" the delta team will will.  

Example:   Show that if f(x) = 7x, then lim as x -> 2 f(x) = 14

Solution:  Let epsilon > 0

Scratch Work:

14 - epsilon < f(x) < 14 + epsilon

for all 2 - delta < x < 2 + delta

or equivalently

14 - epsilon < 7x < 14 + epsilon

or

2 - epsilon/7 < x < 2 + epsilon/7

If we choose delta = epsilon/7

then

2 - epsilon/7 < x < 2 + epsilon/7

implies that

14 - epsilon < 7x < 14 + epsilon

so that

14 - epsilon < f(x) < 14 + epsilon

which proves that the limit is14.

Exercise  

Prove that

A)  if f(x) = 3 - 5x

lim as x -> 4 of f(x) = -17

B)  if f(x) = mx + b is a line then

lim as x -> c of f(x) = mc + b