# Limits via Tables

There's 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) = x2, estimate

.

This is how it'll go. We'll make a table. In one column we'll have values of x, and in the next we'll have the corresponding values of f(x).

First we have x approach 3 from the left.

xf(x)
2.5  6.25
2.77.29
2.97.29
2.98.41
2.998.9401
2.9998.994001

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.

xf(x)
3.1 9.61
3.019.0601
3.0019.006001
3.00019.0006001

The values of f(x) appear to be approaching 9 as x approaches 3 from the right as well. 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.