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.
| 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) 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) |
|---|---|
| 3.1 | 9.61 |
| 3.01 | 9.0601 |
| 3.001 | 9.006001 |
| 3.0001 | 9.0006001 |
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.