Figure

EXAMPLE 1Investigating a Limit Using a Table of Numbers

Investigate $\lim\limits_{x\rightarrow 2}( 2x+5)$ using a table of numbers.

We create Table 1 by evaluating $f(x) =2x+5$ at values of $x$ near 2, choosing numbers $x$ slightly less than 2 and numbers $x$ slightly greater than 2.

TABLE 1

$\underrightarrow{\hbox{numbers $x$ slightly less than 2}}$								$\underleftarrow{\hbox{numbers $x$ slightly greater than 2}}$
$x$	1.99	1.999	1.9999	1.99999	$\rightarrow$	2	$\leftarrow$	2.00001	2.0001	2.001	2.01
$f(x) =2x+5$	8.98	8.998	8.9998	8.99998	$f(x)$ approaches 9			9.00002	9.0002	9.002	9.02

Table 1 suggests that the value of $f(x)=2x+5$ can be made “as close as we please” to 9 by choosing $x$ “sufficiently close” to 2. This suggests that $\lim\limits_{x\rightarrow 2}( 2x+5) =9$ .