Figure 12-3: Oblique asymptote
If the degree of the numerator of a rational function is exactly one more than the degree of the denominator, the graph of the rational function will have an oblique asymptote. An oblique asymptote is neither vertical nor horizontal. The graph of a rational function may have at most one oblique asymptote. It cannot have both a horizontal and an oblique asymptote.
Now, let f(x) be a rational function in which P(x) and Q(x) are polynomial functions:
P(x) anxn + an-1xn-1 + . . . + ao
f(x) = ------ = ---------------------------
Q(x) bmxm + bm-1xm-1 + . . . + bo
When an ≠ 0 and bm ≠ 0 and
n = m + 1, f(x) is an improper rational expression. Using polynomial long
division, it can be converted into the sum of a linear polynomial and a proper
rational expression. That is, f(x) can be written as:
R(x)
f(x) = ax + b + ------
Q(x)
The expression ax + b is the quotient obtained by long division of
P(x) by Q(x). R(x) is the remainder. The degree of
R(x) is less than the degree of Q(x).
Then, because the term R(x)/Q(x) -> 0, as x -> ∞ or x -> -∞, the function f(x) -> ax + b. That is, the line y = ax + b is the oblique asymptote for the graph of f(x).
Example:
x2 + 3x + 4 2
------------- = x + 2 + -------
x + 1 x + 1
Then the equation y = x + 2 is the oblique asymptote for the graph of f(x).
Another example, graph the function:
x2 - 2
f(x) = --------
2x + 4
Using polynomial long division:
1
f(x) = 0.5x - 1 + -------
x + 2
There is a vertical asymptote at x = -2
There are x-axis intercepts at x = -√2, +√2
There is a y-axis intercept at y = -0.5
The oblique asymptote is y = 0.5x - 1
Figure 12-3: Oblique asymptote