###### Example334

Investigate the graph of \(f(x) = \displaystyle{\frac{2}{(x - 3)^2}}\) near \(x = 3\text{.}\)

This function is undefined for \(x = 3\text{,}\) so there is no point on the graph with \(x\)-coordinate \(3\text{.}\) However, we can make a table of values for other values of \(x\text{.}\) Plotting the ordered pairs in the table results in the points shown in Figure335.

\(x\) | \(y\) |

\(0\) | \(\dfrac{2}{9}\) |

\(1\) | \(\dfrac{1}{2}\) |

\(2\) | \(2\) |

\(3\) | undefined |

\(4\) | \(2\) |

\(4\) | \(\dfrac{1}{2}\) |

\(6\) | \(\dfrac{2}{9}\) |

Next, we make a table showing \(x\)-values close to \(3\text{,}\) as in Figure336a. As we choose \(x\)-values closer and closer to \(3\text{,}\) \((x - 3)^2\) gets closer to \(0\text{,}\) so the fraction \(\dfrac{2}{(x-3)^2}\) gets very large. This means that the graph approaches, but never touches, the vertical line \(x=3\text{.}\) In other words, the graph has a vertical asymptote at \(x=3\text{.}\) We indicate the vertical asymptote by a dashed line, as shown in Figure336b.