3.7 Rational functions  (Page 5/16)

Identifying horizontal asymptotes of rational functions

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 $y=0.$

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

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

In this case, the end behavior is $f(x)\approx \frac{3x^2}{x}=3x.$ This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function $g(x)=3x.$ As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of $g(x)=3x$ looks like a diagonal line, and since $f$ will behave similarly to $g,$ it will approach a line close to $y=3x.$ This line is a slant asymptote.

To find the equation of the slant asymptote, divide $\frac{3x^2-2x+1}{x-1}.$ The quotient is $3x+1,$ and the remainder is 2. The slant asymptote is the graph of the line $g(x)=3x+1.$ See [link] .

Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at $y=\frac{a_n}{b_n},$ where $a_n$ and $b_n$ are the leading coefficients of $p\left(x\right)$ and $q\left(x\right)$ for $f(x)=\frac{p(x)}{q(x)},q(x)\ne 0.$

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

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.