1.4 Estimating Limit Values from Tables

Using tables to estimate limit values helps us better visualize what a limit actually is. In order to understand this concept, let’s look at an example. 💭

The easiest way to find a limit is to plug in the given x value into the function. However, you’ll find that the given function isn’t defined at x = 3 because the denominator evaluates to 0. Because we can’t find the function value at x = 3, we find the limit by approaching x = 3. 

1) We have to make a table and the first step is picking the x values to use. Pick a value that's a little bit less than x = 3 (that is, a value that's "to the left" 👈 of 3), so maybe start with something like x = 2.9.

2) Next, add a couple more x-values to your table to simulate the feeling of getting infinitely close to x = 3, from the left.

3) Approach x = 3 from the right just like we did from the left. 

Now that we have our table, we can estimate the limit of f(x) = x - 3 ⁄ x^2 - 9 at x = 3 is 0.1667 or 1 ⁄ 6. 

It is important to pick x values that get infinitely close to the target value. In the example above, we picked numbers that were closer and closer to x = 3. We wouldn’t be able to estimate the limit value if we used numbers with equal increments like 2.25, 2.5, and 2.75.