##### Graphs of Rational Functions - Vertical Asymptotes

If the rational function \(f\) can be written as \(f(x)=\frac{p(x)}{q(x)}\) where \(p\) and \(q\) have no common factors, and \(k\) is a real number solution to the equation \(q(x)=0\text{,}\) then the vertical line \(x=k\) is a vertical asymptote on a graph of \(y=f(x)\text{.}\)

The vertical asymptote is not actually part of the function, it is simply a visual aid to help the viewer of the graph understand that from either side of the line \(x=k\text{,}\) the actual function is actual soaring up and up without limit or plummeting down and down without limit.

The function \(f(x)=\frac{2}{x-3}\) is shown in FigureĀ 12.9.1. The vertical asymptote \(x=3\) is also shown. From the left of the line, the actual function value falls without limit. We communicate this fall using the limit equation

The limit equation stated above is read aloud as "the limit of \(f(x)\text{,}\) as \(x\) approaches 3 from the left, is (or equals) negative infinity." Similarly, for the function shown in FigureĀ 12.9.1 we would say that "the limit of \(f(x)\text{,}\) as \(x\) approaches 3 from the right, is infinity."" In symbols:

We see in TableĀ 12.9.2 the cause of the asymptotic behavior. As the value of \(x\) approaches 3 from either side of 3, the absolute value of the expression \(x-3\) gets closer and closer to zero which causes the absolute value of \(\frac{2}{x-3}\) to grow and grow with limit.

\(x\) | \(x-3\) | \(\frac{2}{x-3}\) |

\(2.9\) | \(-0.1\) | \(-20\) |

\(2.99\) | \(-0.01\) | \(-200\) |

\(2.999\) | \(-0.001\) | \(-2,000\) |

\(3.001\) | \(0.001\) | \(2,000\) |

\(3.01\) | \(0.01\) | \(200\) |

\(3.1\) | \(0.1\) | \(20\) |

###### Example12.9.3

Determine the vertical asymptotes for the function

Also, use limit notation to express the behavior of \(f\) from either side of both asymptotes.

Setting the denominator equal to zero and solving we get the following.

The vertical asymptotes are the lines \(x=-5\) and \(x=-2\text{.}\) We can determine whether the function soars upward or plunges to the depths by testing values of \(x\) a tenth of a unit to the left and right of each asymptote. Specifically, we are only interested in the signs of each of the factors in \(f\) from wish we can infer the over all sign for \(f\text{.}\) This is down in TableĀ 12.9.4.

\(x\) | \(x-4\) | \(x+5\) | \(x+2\) | \(\frac{x-4}{(x+5)(x+2)}\) | Implied Limit |

\(-5.1\) | \(-\) | \(-\) | \(-\) | \(-\) | \(\lim_{x \to -5^-}f(x)=-\infty\) |

\(-4.9\) | \(-\) | \(+\) | \(-\) | \(+\) | \(\lim_{x \to -5^+}f(x)=\infty\) |

\(-2.1\) | \(-\) | \(+\) | \(-\) | \(+\) | \(\lim_{x \to -2^-}f(x)=\infty\) |

\(-1.9\) | \(-\) | \(+\) | \(+\) | \(-\) | \(\lim_{x \to -2^+}f(x)=-\infty\) |

From the sign analysis in TableĀ 12.9.4 we can infer the following for limits.

As a bonus, a graph of the function \(y=\frac{x-4}{(x+5)(x+2)}\) is shown in FigureĀ 12.9.5.