Investigating a Limit Using a Graph

Use a graph to investigate \(\lim\limits_{x\rightarrow 0}f(x)\) if \(f( x) =\left\{ \begin{array}{l@{\qquad}l} x & \hbox{if }\quad x<0 \\ 1 & \hbox{if }\quad x>0 \end{array} \right. .\)

Solution Figure 13 shows the graph of \(f\). We first investigate the one-sided limits. The graph suggests that, as \(x\) approaches 0 from the left, \[ \begin{equation*} \lim_{x\rightarrow 0^{-}}f(x)=0 \end{equation*} \]

and as \(x\) approaches 0 from the right, \[ \begin{equation*} \lim_{x\rightarrow 0^{+}}f(x)=1 \end{equation*} \]

Since there is no single number that the values of \(f\) approach when \(x\) is close to 0, we conclude that \(\lim\limits_{x\rightarrow 0}f(x)\) does not exist.