Investigating a Limit Using a Table of Numbers

Investigate \(\lim\limits_{x\rightarrow 0}\dfrac{\sin x}{x}\) using a table of numbers.

Solution The domain of the function \(f(x)=\dfrac{\sin x}{x}\) is \(\{x~|~x\neq 0\}\). So, \(f\) is defined everywhere in an open interval containing 0, except for 0.

We create Table 3, by investigating one-sided limits of \(\dfrac{\sin x}{x}\) as \(x\) approaches 0, choosing numbers (in radians) slightly less than 0 and numbers slightly greater than 0.

TABLE 3
\(\underrightarrow{x~\hbox{approaches 0 from the left}}\) \(\underleftarrow{x~\hbox{approaches 0 from the right}}\)
\(x\) (in radians) -0.02 -0.01 0.005 \(\rightarrow\) 0 \(\leftarrow\) 0.005 0.01 0.02
\(f(x) =\dfrac{\sin x}{x}\) 0.99993 0.99998 0.999996 \(f(x)\) approaches 1 0.999996 0.99998 0.99993

Table 3 suggests that \(\lim\limits_{x\rightarrow 0^{-}}f(x)=1\) and \(\lim\limits_{x\rightarrow 0^{+}}f(x)=1\). This suggests that \(\lim\limits_{x\rightarrow 0}\dfrac{\sin x}{x}=1\).