While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.

There are three distinct outcomes when checking for horizontal asymptotes:

Case 1: If the degree of the denominator > degree of the numerator, there is a horizontal asymptote at [latex]y=0[/latex].

Example: [latex]f\left(x\right)=\dfrac{4x+2}{{x}^{2}+4x - 5}[/latex]

In this case the end behavior is [latex]f\left(x\right)\approx \frac{4x}{{x}^{2}}=\frac{4}{x}[/latex]. This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\left(x\right)=\frac{4}{x}[/latex], and the outputs will approach zero, resulting in a horizontal asymptote at [latex]y=0[/latex]. Note that this graph crosses the horizontal asymptote.

Horizontal Asymptote [latex]y=0[/latex] when [latex]f\left(x\right)=\dfrac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne{0}\text{ where degree of }p<\text{degree of q}[/latex].

Case 2: If the degree of the denominator < degree of the numerator by one, we get a slant asymptote.

Example: [latex]f\left(x\right)=\dfrac{3{x}^{2}-2x+1}{x - 1}[/latex]

In this case the end behavior is [latex]f\left(x\right)\approx \frac{3{x}^{2}}{x}=3x[/latex]. This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\left(x\right)=3x[/latex]. As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of [latex]g\left(x\right)=3x[/latex] looks like a diagonal line, and since [latex]f[/latex] will behave similarly to [latex]g[/latex], it will approach a line close to [latex]y=3x[/latex]. This line is a slant asymptote (NOTE: the graph of the function itself is just a “sketch” within the parameters of the asymptotes).

Slant Asymptote when [latex]f\left(x\right)=\dfrac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0[/latex] where degree of [latex]p>\text{ degree of }q\text{ by }1[/latex].

To find the equation of the slant asymptote, divide [latex]\dfrac{3{x}^{2}-2x+1}{x - 1}[/latex]. The quotient is [latex]3x+1[/latex], and the remainder is 2. The slant asymptote is the graph of the line [latex]g\left(x\right)=3x+1[/latex].

Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at [latex]y=\frac{{a}_{n}}{{b}_{n}}[/latex], where [latex]{a}_{n}[/latex] and [latex]{b}_{n}[/latex] are the leading coefficients of [latex]p\left(x\right)[/latex] and [latex]q\left(x\right)[/latex] for [latex]f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0[/latex].

Example: [latex]f\left(x\right)=\dfrac{3{x}^{2}+2}{{x}^{2}+4x - 5}[/latex]

In this case the end behavior is [latex]f\left(x\right)\approx \frac{3{x}^{2}}{{x}^{2}}=3[/latex]. This tells us that as the inputs grow large, this function will behave like the function [latex]g\left(x\right)=3[/latex], which is a horizontal line. As [latex]x\to \pm \infty ,f\left(x\right)\to 3[/latex], resulting in a horizontal asymptote at [latex]y=3[/latex]. Note that this graph crosses the horizontal asymptote.

Horizontal Asymptote when [latex]f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0\text{ where degree of }p=\text{degree of }q[/latex].

Notice that, while the graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.

It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction. For instance, if we had the function

[latex]f\left(x\right)=\dfrac{3{x}^{5}-{x}^{2}}{x+3}[/latex]

with end behavior

[latex]f\left(x\right)\approx \dfrac{3{x}^{5}}{x}=3{x}^{4}[/latex],

the end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient.

As [latex]x\to \pm \infty , f\left(x\right)\to \infty [/latex]

Watch this video to see more worked examples of determining which kind of horizontal asymptote a rational function will have.

Example: Identifying Horizontal and Vertical Asymptotes

Find the horizontal and vertical asymptotes of the function

[latex]f\left(x\right)=\dfrac{\left(x - 2\right)\left(x+3\right)}{\left(x - 1\right)\left(x+2\right)\left(x - 5\right)}[/latex]

Show Solution

First, note that this function has no common factors, so there are no potential removable discontinuities.

The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The denominator will be zero at [latex]x=1,-2,\text{and }5[/latex], indicating vertical asymptotes at these values.

The numerator has degree 2, while the denominator has degree 3. Since the degree of the denominator is greater than the degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards zero as the inputs get large, and so as [latex]x\to \pm \infty , f\left(x\right)\to 0[/latex]. This function will have a horizontal asymptote at [latex]y=0[/latex].

Example: Finding the Intercepts of a Rational Function

Find the intercepts of [latex]f\left(x\right)=\dfrac{\left(x - 2\right)\left(x+3\right)}{\left(x - 1\right)\left(x+2\right)\left(x - 5\right)}[/latex].

Show Solution

We can find the [latex]y[/latex]-intercept by evaluating the function at zero

[latex]\begin{align}f\left(0\right)&=\dfrac{\left(0 - 2\right)\left(0+3\right)}{\left(0 - 1\right)\left(0+2\right)\left(0 - 5\right)} \\[1mm] &=\frac{-6}{10} \\[1mm] &=-\frac{3}{5} \\[1mm] &=-0.6 \end{align}[/latex]

The [latex]x[/latex]-intercepts will occur when the function is equal to zero. A rational function is equal to 0 when the numerator is 0, as long as the denominator is not also 0.

[latex]\begin{align} 0&=\frac{\left(x - 2\right)\left(x+3\right)}{\left(x - 1\right)\left(x+2\right)\left(x - 5\right)} \\[1mm] 0&=\left(x - 2\right)\left(x+3\right)\end{align}[/latex]

[latex]x=2, -3[/latex]

The [latex]y[/latex]-intercept is [latex]\left(0,-0.6\right)[/latex], the [latex]x[/latex]-intercepts are [latex]\left(2,0\right)[/latex] and [latex]\left(-3,0\right)[/latex].

Watch the following video to see more worked examples of finding asymptotes, intercepts and holes of rational functions.

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