Horizontal Asymptotes of Rational Functions

What is a Horizontal Asymptote?
A horizontal asymptote is a horizontal line \(y = b\) where the graph of a function \(f(x)\) approaches \(b\) as \(x\) approaches positive infinity (\(x \to \infty\)) or negative infinity (\(x \to -\infty\)). Horizontal asymptotes describe the end behavior of a function as \(x\) gets very large or very small.

The following diagrams show how to find the horizontal asymptotes of rational functions. Scroll down the page for more examples and solutions on how to find horizontal asymptotes.

 

How to find Horizontal Asymptotes?
We can find the horizontal asymptotes of a rational function \(f(x) = \frac{P(x)}{Q(x)}\) by comparing the degrees of the numerator \(n\) and the denominator \(m\):

1. If \(n < m\): The horizontal asymptote is \(y = 0\) (the x-axis).
2. If \(n = m\): The horizontal asymptote is \(y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}\).
3. If \(n > m\): There is no horizontal asymptote. Instead, there might be a slant (or oblique) asymptote.

For other types of functions (e.g., exponential, logarithmic), horizontal asymptotes are found by evaluating the limits as \(x \to \infty\) and \(x \to -\infty\).

Graphing Rational Functions, n less than m
There are different characteristics to look for when graphing rational functions. When graphing rational functions where the degree of the numerator function is less than the degree of denominator function, we know that y = 0 is a horizontal asymptote. When the degree of the numerator is equal to or greater than that of the denominator, there are other techniques for graphing rational functions.

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Graphing Rational Functions, n = m
There are different characteristics to look for when creating rational function graphs. With rational function graphs where the degree of the numerator function is equal to the degree of denominator function, we can find a horizontal asymptote. When the degree of the numerator is less than or greater than that of the denominator, there are other techniques for drawing rational function graphs.
How to recognize when a rational function has a horizontal asymptote, and how to find its equation.

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Graphing Rational Functions, n > m
There are different characteristics to look for when creating rational function graphs. With rational function graphs where the degree of the numerator function is greater than the degree of denominator function, there is no horizontal asymptote, but there may be an oblique asymptote.

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