2.2 The limit of a function

Problem-solving strategy: evaluating a limit using a table of functional values

1. To evaluate $\underset{x\to a}{\text{lim}}f\left(x\right),$ we begin by completing a table of functional values. We should choose two sets of x -values—one set of values approaching a and less than a , and another set of values approaching a and greater than a . [link] demonstrates what your tables might look like.
   

   Table of functional values for $\underset{x\to a}{\text{lim}}f\left(x\right)$

   x	$f\left(x\right)$		x	$f\left(x\right)$
   $a-0.1$	$f\left(a-0.1\right)$		$a+0.1$	$f\left(a+0.1\right)$
   $a-0.01$	$f\left(a-0.01\right)$	$a+0.01$	$f\left(a+0.01\right)$
   $a-0.001$	$f\left(a-0.001\right)$	$a+0.001$	$f\left(a+0.001\right)$
   $a-0.0001$	$f\left(a-0.0001\right)$	$a+0.0001$	$f\left(a+0.0001\right)$
   Use additional values as necessary.	Use additional values as necessary.

2. Next, let’s look at the values in each of the $f\left(x\right)$ columns and determine whether the values seem to be approaching a single value as we move down each column. In our columns, we look at the sequence $f\left(a-0.1\right),f\left(a-0.01\right),f\left(a-0.001\right).,f\left(a-0.0001\right),$ and so on, and $f\left(a+0.1\right),f\left(a+0.01\right),f\left(a+0.001\right),f\left(a+0.0001\right),$ and so on. ( Note : Although we have chosen the x -values $a\pm 0.1,a\pm 0.01,a\pm 0.001,a\pm 0.0001,$ and so forth, and these values will probably work nearly every time, on very rare occasions we may need to modify our choices.)
3. If both columns approach a common y -value L , we state $\underset{x\to a}{\text{lim}}f\left(x\right)=L.$ We can use the following strategy to confirm the result obtained from the table or as an alternative method for estimating a limit.
4. Using a graphing calculator or computer software that allows us graph functions, we can plot the function $f\left(x\right),$ making sure the functional values of $f\left(x\right)$ for x -values near a are in our window. We can use the trace feature to move along the graph of the function and watch the y -value readout as the x -values approach a . If the y -values approach L as our x -values approach a from both directions, then $\underset{x\to a}{\text{lim}}f\left(x\right)=L.$ We may need to zoom in on our graph and repeat this process several times.