Category Archives: Table

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.

f(x) = {x^3+x^2-4x-4}/{x+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 f(x) right -3as x right -1 from the left would be written.

lim{x right -1^{-}}{f(x)}=-3

x-.9-.99-.999-.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 f(x) right -3as x right -1 from the right would be written.

lim{x right -1^{+}}{f(x)}=-3

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: lim{x right c}{f(x)}=L

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

lim{x right -1}{f(x)}=-3