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Using Tables for Estimation

Previously, we discussed and practiced using graphs to estimate limits. However, graphs are not always available or easy to use. In such cases, one alternative is to construct a table of values and use it to estimate the limit.

Consider some function that you want to find the limit of as approaches . We can sample the function at values of that are close to and use the values to estimate the limit.

Our samples can be: , , , , ,

Let's say that we find the following values for :

From this table, we can see that as gets closer to , the value of gets closer to . Hence, we can say with reasonable confidence that:

This method is particularly useful when you have a function that is difficult to graph or when you need a more precise estimate than what a graph can provide.

Double-Sided Tables​

Recall that limits can sometimes be different depending on whether you approach from the left or the right. This can also be reflected in the table of values. Consider the following sample values for as approaches from the left and the right:

Notice that as approaches from the left, the value of gets closer to .

On the other hand, as approaches from the right, the value of gets closer to .

Hence, we can construct two one-sided limits from this table:

Precautions and Limitations​

With this method, there are some things to keep in mind to ensure accurate estimates:

  1. Sampling - we want to sample values that are close to the limit point but not exactly at the limit point. This is because we are trying to estimate the limit as approaches a certain value, not the value of the function at that point (which may not even exist).
  2. Accuracy - the accuracy of the estimate depends on the number of samples and how close they are to the limit point. More samples and closer samples will generally give a better estimate.

When you get the results from the table;

  1. The values for should get closer to a single value as approaches the limit point.
  2. If the limit is two-sided (i.e., same from both sides), the values from both sides should get closer to the same value.
  3. If the limit is different from both sides, the values from each side should get closer to their respective values.

Summary and Next Steps​

In this section, we introduced the concept of using tables to estimate limits.

Here are the key points to remember:

  1. Tables can be used to estimate limits when graphs are not available or not practical.
  2. Sampling values close to the limit point can help estimate the limit.
  3. Double-sided tables can be used to estimate one-sided limits.
  4. Accuracy of the estimate depends on the number of samples and how close they are to the limit point.

We've now covered two methods for estimating limits: using graphs and using tables, but what about actually computing limits? In the next section, we will look at some techniques for computing limits analytically, starting with the various algebraic properties of limits.