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Functions, Graphs, and Limits

Functions, Graphs, and Limits

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.

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