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 $\lim\limits_{x\rightarrow 2}f(x)=7$.