If the graph of a rational function approaches a horizontal line as the values of x assume increasingly large values, both positive and negative, the graph is said to have a horizontal asymptote. Some rational functions have a horizontal asymptote, but not all.
The graphs of the functions shown above do not fully show the behavior of f(x) for very large x, but it is not surprising to be told that f(x) = and each have y = 0 as a horizontal asymptote , and has y = 1 as a horizontal asymptote. To find a horizontal asymptote, we recall that as x becomes very large in magnitude, either positive or negative, then becomes very small in magnitude. Press the button to see the graph of
In mathematical language, as
This means that when x is extremely large, and in fact all negative powers of x can be ignored. Therefore, to find out how the function f(x) behaves for large values of x, we divide each term in both the denominator and the numerator by the highest power of x in the denominator and determine what happens when x is large enough to ignore negative powers of x.
Thus in the case of , we have which approaches the line = 1 as . Therefore this function has a horizontal asymptote, y = 1. Click the button to see the graph of
In the case of , we have which approaches the line as , so that y = 0 is a horizontal asymptote. Click on the button to see the graph of
This procedure leads to the following general result: * 1 * If the degree of the denominator exceeds that of the numerator, y = 0 is a horizontal asymptote;
* 2 * If the leading coefficient of the numerator is a and the leading coefficient of the denominator is b and their degrees are equal then is a horizontal asymptote;
* 3 * If the degree of the numerator exceeds that of the denominator, there is no horizontal asymptote.
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