Investigating a Limit Using a Graph

Use a graph to investigate \(\lim\limits_{x\rightarrow 2}f( x)\) if \(f(x)=\left\{ \begin{array}{c@{\qquad}l} 3x+1 & \hbox{if }\quad x\neq 2 \\ 10 & \hbox{if }\quad x=2 \end{array} \right. . \)

Solution The function \(f\) is a piecewise-defined function. Its graph is shown in Figure 12. Observe that as \(x\) approaches 2 from the left, the value of \(f\) is close to 7, and as \(x\) approaches 2 from the right, the value of \(f\) is close to 7. In fact, we can make the value of \(f\) as close as we please to 7 by choosing \(x\) sufficiently close to 2 but not equal to 2. This suggests \(\lim\limits_{x\rightarrow 2}f(x)=7\).