Definition of Vertical and Horizontal Asymptotes

1. The line x = a is a vertical asymptote of the graph of f if       or

     - as x  a, either from the right or from the left.

2. The line y = b is a horizontal asymptote of the graph of f if    as    

      or  -.

The line x = 0 is a vertical asymptote of the graph of f, shown below.

The graph of f also has a horizontal asymptote - the line y = 0.  This means the values of f(x) = 1/x approach zero as x increases or decreases without bound.

    as   -                                                             as   

       approaches 0 as x                                                                                          approaches 0 as x

       decreases without bound.                                                                                     increases without bound.

  Asymptotes of a Rational Function

Let f be the rational function

 

           

 where N(x) and D(x) have no common factors.

1. The graph of f has vertical asymptotes at the zeros of D(x).

2. The graph of f has at most one horizontal asymptote determined by comparing the degrees

    of N(x) and D(x).

            a. If n > m, the line y = 0 (the x-axis) is a horizontal asymptote.

            b. If n = m, the line y =   is a horizontal asymptote.

            c. If n > m, the graph of f has no horizontal asymptote.  

 

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