Limits a Geometric and Numeric Approach

Limits Using Tables

Consider the function

x2 - 1
f(x)  =
x + 1

Notice that this function is undefined at x = -1. In calculus undefined is not as precise as possible.  Instead one asks, what does the y value "look like" when the x value is near -1.  The table below demonstrates:

 x -0.9 -1.1 -0.99 -1.01 -0.999 -1.001 y -1.9 -2.1 -1.99 -2.01 -1.999 -2.001

We see that if x is close to -1, then f(x) is close to -2.  We say "The limit of f(x) as x approaches -1 is -2" and write

If the y value does not tend toward a single number as x tends towards a, then we say that the limit does not exist as x approaches a.

Exercise

Use a table to find the following limit if it exists.

Looking at the graph of a function is another convenient way of determining a limit.  For example, a computer was used to graph the the function

x2 - 1
f(x)  =
x + 1

Notice that the computer indicates that the y value approaches -2 as the x value approaches -1.  In fact, the computer ignores the fact that the function is undefined at x = -1.

Exercise:

Use a graphing calculator to find the limits from the prior exercise if they exist.

The Epsilon-Delta Game

Choose a function and a number.  Let partner A select a y range.  Partner B must find an x range so that the graph leaves the box on the sides and not the top and bottom.  If Partner B can always win, then the function has a limit at that number.  We will play this game first with me as Partner B and the class as Partner A.