有理函数：求水平渐近线

As we know, a rational function $r(x)$ is the quotient of two polynomial functions. A horizontal asymptote of a function is a horizontal line that a functions approaches, but never touches. Basically, the function reaches that value at $\pm \infty$. This problem is then reduced to finding $\lim_{x \to \pm\infty} r(x)$, which can be done like this. If the limit is not $\pm\infty$, then the function has a horizontal asymptote at that value.

(Video) How to Find the Horizontal Asymptote (NancyPi)

by NancyPi

Luckily for us, Nancy has 3 great rules to help us find the horizontal asymptote. These rules are a way for us to avoid using limits, as those can be harder. Instead, you must examine the degree of the polynomial.

Rule #1: If the top degree is less than the bottom degree.

Nancy's example: $$\frac{x+1}{x^2-1}$$

Since the degree on top is smaller, the horizontal asymptote is $y=0$.

Rule #2: If the top degree equals the bottom degree.

Nancy's example: $$\frac{x^2+1}{3x^2+1}$$

Since the degrees on the top and bottom are equal, divide the highest degree terms in the numerator and denominator.

$$\frac{x^2}{3x^2} = \frac{1}{3}$$

So the horizontal asymptote is $y=\frac{1}{3}$

Rule #3: If the top degree is greater than the bottom degree.

Nancy's example: $$\frac{x^2+1}{x-3}$$

There is actually no horizontal asymptote. There is an oblique asymptote, but you will learn about that later. For now, just remember that there is no horizontal asymptote.

Horizontal Asymptotes of Rational Functions

Now that we've covered vertical asymptotes, we can move on to the next type.

In this lesson, we are going to learn about horizontal asymptotes. A horizontal asymptote is an invisible flat, straight line that a graph gets close to, but never actually touches.

We usually see horizontal asymptotes in rational functions, which are functions with polynomials on the top and bottom. A rational function might look like this: $f(x)=\frac1x$. As $x$ gets really big, the value of $\frac1x$ gets really small. It gets super close to $0$, but never actually becomes $0$. That means the horizontal asymptote is $y=0$.

To find horizontal asymptotes, we look at the highest powers of $x$ in the top and bottom of the fraction. If the top power is smaller than the bottom, the asymptote is $y=0$. If the top and bottom powers are the same, divide the leading coefficients.

For example, if we have: $f(x) = \frac{2x^3+3}{x^2+1}$

Then the asymptote is $y=2$ because $2$ and $1$ are the leading coefficients.

Horizontal asymptotes show us what happens far to the left or right of the graph. They help us to understand how functions behave as numbers get really big or really small.

Rational Functions: Horiztonal Asymptotes

A rational function is a fraction where both the numerator and denominator are polynomials. When you graph a rational function, the curve often gets closer to a line as $x$ becomes very large or very small. This line is called a horizontal asymptote.

A horizontal asymptote shows the value that the function approaches as $x \rightarrow \infty$ or $x \rightarrow -\infty$. It helps you understand the end behavior of the graph.

To find the horizontal asymptote, compare the degrees (highest powers of $x$) in the numerator and denominator.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y=0$

• If the degrees are equal, dividing the leading coefficients. For example: $$\frac{3x^2}{2x^2-5} \rightarrow y=\frac{3}{2}$$

• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. However - there may be a slant asymptote.

Example

Let's think through an example together.

$$\frac{x}{x+1}$$

The degree of the numerator and the degree of the denominator are equal. Therefore, the asymptote is $y=1$.

Horizontal asymptotes help you sketch graphs and understand how functions behave far away from the center. Even though the graph gets close to the asymptote, it may still cross it sometimes.