PreCalculus

12.2. Horizontal asymptotes

Given the rational function in which P(x) and Q(x) are polynomial functions:

        P(x)     a_nxⁿ + a_n-1x^n-1 + . . . + a_o
f(x) = ------ = ---------------------------
        Q(x)     b_mx^m + b_m-1x^m-1 + . . . + b_o

The horizontal asymptote is the limiting value of f(x) as x -> ∞ or x -> -∞. There are three possibilities:

If n > m there is no horizontal asymptote, f(x) is unbounded.
If n < m then the horizontal asymptote is the x-axis: y = 0
If n = m then the horizontal asymptote is: y = a_n/b_m

An example, find the horizontal asymptote of:

        2
f(x) = ---
        x

lim f(x) = lim 2/x = 2/∞ = 0
^{x -> ∞}       ^{x -> ∞}

That is, since n < m, the horizontal asymptote is given by: y = 0.

Another example, graph the function:

          2x²
f(x) = --------
        x² + 2

                    2          2
lim f(x) = lim --------- = --------- = 2
^{x -> ∞}      ^{x -> ∞}  1 + 2/x²    1 + 2/∞

That is, since n = m, the horizontal asymptote is given by: y = a_n/b_m = 2/1 = 2.

The graph is symmetrical about the y-axis because it is unchanged when x is replaced by -x:

         2(-x)²       2x² 
f(-x) = --------- = -------
        (-x)² + 2    x² + 2

The x-intercept is found by solving:

0 = 2x²/(x² + 2)

x = 0

The y-intercept is

f(0) = 2×0²/(0² + 2) = 0

Figure 12-2: Horizontal asymptote