# Functions, Graphs, and Limits

# Limits via Tables

There is more than one way to approach (pun absolutely intended) limit problems. We've already looked at graphs and equations.

Another way to estimate the limit of a function is to use a calculator to see what the function approaches as we plug in values of *x* that get closer and closer to some value *a*. To keep things organized, now we'll use tables to get the lowdown on functions.

### Sample Problem

If *f*(*x*) = *x*^{2}, estimate

.

We make a table. In one column we'll have values of *x*, and in the next we'll have the corresponding values of *f*.

First we have *x* approach 3 from the left.

The values of *f*(*x*) in the table appear to be getting closer to 9 as *x* approaches 3 from the left. We'll see what happens if *x* approaches 3 from the right.

x | f(x) |
---|---|

2.5 | 6.25 |

2.7 | 7.29 |

2.9 | 7.29 |

2.9 | 8.41 |

2.99 | 8.9401 |

2.999 | 8.994001 |

The values of *f*(*x*) appear to be approaching 9 as *x* approaches 3 from the right, also. We can now shout from the rooftops that, indeed, = 9.

When using tables to determine limits, there's no particular rule about what numbers to plug in for *x* as it approaches a number *a*. As long as we look at lots of values of *x*, and let them get really close (as in, 0.00001 close) to *a*, we should be fine.

x | f(x) |
---|---|

3.1 | 9.61 |

3.01 | 9.0601 |

3.001 | 9.006001 |

3.0001 | 9.0006001 |