# Limit at a Hole

The function f below is undefined for x=-1. Using a table, analyze what is happening to f(x) as x approaches -1.

Filling out this table will help us decide what is happening to f(x) as x is getting closer to -1 from the left. (x’s that are smaller then -1)

 x -1.1 -1.01 -1.001 -1.0001 -1 f(x) ?

Filling out this table will help us decide what is happening to f(x) as x is getting closer to -1 from the right. (x’s that are larger then -1)

 x -.9 -.99 -.999 -.9999 -1 f(x) ?

Here is a video that will help you use the features in your calculator to fill in the values of the table:

 x -1.1 -1.01 -1.001 -1.0001 -1 f(x) -2.79 -2.9799 -2.998 -2.9998 ?

As you can see from the table, f(x) approaches -3 as x approaches -1 from the left. Symbolically as from the left would be written.

 x -0.9 -0.99 -0.999 -0.9999 -1 f(x) -3.19 -3.0199 -3.002 -3.0002 ?

As you can see from the table, f(x) approaches -3 as x approaches -1 from the right. Symbolically as from the right would be written.

These are called one sided limits.

Informal Definition of a Limit: If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, then the limit of f(x) , as x approaches c is L.

In general:

In our example, f(x) becomes arbitrarily close to -3 as x approaches -1 from the left and right.