Investigating a Limit Using a Table of Numbers

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

Solution 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 \(\lim\limits_{x\rightarrow 2}( 2x+5) =9\).