# How to find the horizontal asymptote of a rational function?

Mathematics is a very complex subject and often even studying with a certain commitment we are not able to perfectly understand some topics. In these cases, we will be able to search for guides on some websites that with some examples will explain the subject or formulas. In this guide, we see specifically the way to find the horizontal asymptote of a rational function.

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## How to find the horizontal asymptote of a rational function?

A horizontal asymptote of a rational function is a horizontal line that the graph of the function approaches as the x-value gets larger or smaller.

The horizontal asymptote can be found by looking at the degrees of the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, the horizontal asymptote is **y=a**, where a is the leading coefficient of the numerator.

And if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. If the degrees of the numerator and denominator are equal, the horizontal asymptote is y=a. Where a is the quotient of the leading coefficients of the numerator and denominator.

For example, the function **f(x)=2x+3** has a degree of 1 in the numerator and a degree of 0 in the denominator. So the horizontal asymptote is y=2.

To find the horizontal asymptote of a rational function, divide the leading coefficients of the numerator and denominator.

- For example, if the function is f(x) = (2x+3)/(x-5), then the horizontal asymptote is f(x) = 2/1 = 2.
- Another example, if the function is f(x) = (4x+1)/(2x-3), then the horizontal asymptote is f(x) = 2/1 = 2.
- In general, if the function is f(x) = (ax+b)/(cx+d), then the horizontal asymptote is f(x) = a/c.

### Another way

Another typical example of a function having a horizontal asymptote can be the one we are going to define with a formula and which will then allow us to display it in a graph. Specifically we write the **following: f (x) = 3x ^ 2 + 4x + 9 / x ^ 2 + 2.** If we calculate the limit using the aforementioned known rules of the calculation of limits for x which tends to be more infinite or less infinite, we will obtain as final value 3.

This result has its own logic since **3x ^ 2 and x ^ 2** are infinite of the same order and, therefore, the final limit can be calculated as the ratio of the respective coefficients (in this case) **3/1 = 3.** When we go to draw our graph, the horizontal line passing through the value of the ordinate 3 will constitute our horizontal asymptote.

## What is the formula for horizontal asymptote?

A horizontal asymptote of a function is a line that the function approaches as it goes towards infinity or negative infinity.

There are a few different formulas that can be used to find a horizontal asymptote, but one of the most commonly used is: **y = a * x + b**

Where “a” is the slope of the asymptote and “b” is the y-intercept.

Another way to find a horizontal asymptote is to look at the highest power of the x-term in the function. If the highest power is even, then the asymptote will be at **y = 0**. If the highest power is odd, then the asymptote will be at **y = infinity or y = -infinity.**