## Horizontal Asymptote Calculator

Horizontal asymptote are known as the horizontal lines. Here the horizontal refers to the degree of x-axis, where the denominator will be higher than the numerator. Make use of the below online analytic geometry calculator which is used to find the horizontal asymptote point by entering your rational expressions/functions. On submitting your values you will be able to see the result along with the graph.

### Find Asymptote Point of a Rational Function / Equation in Graph

Horizontal asymptote are known as the horizontal lines. Here the horizontal refers to the degree of x-axis, where the denominator will be higher than the numerator. Make use of the below online analytic geometry calculator which is used to find the horizontal asymptote point by entering your rational expressions/functions. On submitting your values you will be able to see the result along with the graph.

#### Formula:

Method 1:The line y = L is called a Horizontal asymptote of the curve y = f(x) if either

Method 2:

For the rational function, f(x)

In equation of Horizontal Asymptotes,

1. If the degree of x in the numerator is less than the degree of x in the denominator then y = 0 is the Horizontal asymptote. 2. If the degree of x in the numerator is equal to the degree of x in the denominator then y = c where c is obtained by dividing the leading coefficients.

### Example:

Consider a polynomial equation : 2x^2+4x+1 / x^2-16

#### Step 1 :

Numerator's highest degree polynomial is 2 and denominator's highest degree polynomial is 2. Both the denominator and numerator have the same highest degree polynomials, we divide the coefficients of higher degree polynomials.

= 2 / 1

y = 2 is the horizontal asymptote.

## Asymptote Calculator

A straight line is called an asymptote to the curve y = f (x) if, in layman’s term, the curve touches the line at infinity.

### What is Asymptote Calculator?

'Cuemath's Asymptote Calculator' is an online tool that helps to calculate the asymptotic graph for a given function. Cuemath's Asymptote Calculator helps you to find an asymptotic graph for a given function within a few seconds.

### How to Use Asymptote Calculator?

Please follow the steps below on how to use the calculator:

**Step1:**Enter the function with respect to one variable in the given input boxes.**Step 2:**Click on the**"Compute"**button to find an asymptotic graph for a given function**Step 3:**Click on the**"Reset"**button to clear the fields and find the asymptotic graph for different functions.

### How to Find Asymptotes?

An asymptote is defined as a line being approached by a curve but doesn't meet it infinitely or you can say that asymptote is a line to which the curve converges. The asymptote never crosses the curve even though they get infinitely close.

There are three types of asymptotes: 1.Horizontal asymptote 2.Vertical asymptote 3.Slant asymptote

**1.Horizontal asymptote: **The method to find the horizontal asymptote changes based on the degrees of the polynomials in the numerator and denominator of the function.

- If both the polynomials have the same degree, divide the coefficients of the largest degree terms. This is your asymptote!
- If the degree of the numerator is less than the denominator, then the asymptote is located at y=0.
- If the degree of the numerator is greater than the denominator, then there is no horizontal asymptote!

**2.Vertical asymptote:** A vertical asymptote occurs in rational functions at the points when the denominator is zero and the numerator is not equal to zero. We can find the vertical asymptote by equating the denominator of the rational function to zero.

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**Solved Example:**

Find asymptote of given function f(x) = (x + 5) / (x - 3)

**Solution : **

To find a vertical asymptote, equate the denominator of the rational function to zero.

x - 3 = 0

x = 3

So, there exists a vertical asymptote at x = 3

\(\lim _{x \rightarrow 3+} f(x)=\pm \infty, \quad \lim _{x \rightarrow 3-} f(x)=\pm \infty\)

In this case, we have the horizontal asymptote at the point y=1 as it falls under case -1. (numerator and denominator are of same degree: linear)

Similarly, you can try the calculator and find the asymptotes for the following:

- (x
^{2}+ x +1) / (x + 1) - (x + 5) / ( x - 6)

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### Most Used Actions

\mathrm{domain} | \mathrm{range} | \mathrm{inverse} | \mathrm{extreme\:points} | \mathrm{asymptotes} |

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### Related

### Number Line

### Graph

### Examples

function-asymptotes-calculator

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## Asymptote Calculator

### Solution

**Your input: find the vertical, horizontal and slant asymptotes of the function $$$f(x)=\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}$$$**

### Vertical Asymptotes

The line $$$x=L$$$ is a vertical asymptote of the function $$$y=\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}$$$, if the limit of the function (one-sided) at this point is infinite.

In other words, it means that possible points are points where the denominator equals $$$0$$$ or doesn't exist.

So, find the points where the denominator equals $$$0$$$ and check them.

$$$x=-5$$$, check:

$$$\lim_{x \to -5^+}\left(\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{\left(x + 3\right) \left(x + 5\right)}\right)=-\infty$$$ (for steps, see limit calculator).

Since the limit is infinite, then $$$x=-5$$$ is a vertical asymptote.

$$$x=-3$$$, check:

$$$\lim_{x \to -3^+}\left(\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{\left(x + 3\right) \left(x + 5\right)}\right)=\infty$$$ (for steps, see limit calculator).

Since the limit is infinite, then $$$x=-3$$$ is a vertical asymptote.

### Horizontal Asymptotes

Line $$$y=L$$$ is a horizontal asymptote of the function $$$y=f{\left(x \right)}$$$, if either $$$\lim_{x \to \infty} f{\left(x \right)}=L$$$ or $$$\lim_{x \to -\infty} f{\left(x \right)}=L$$$, and $$$L$$$ is finite.

Calculate the limits:

$$$\lim_{x \to \infty}\left(\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}\right)=\infty$$$ (for steps, see limit calculator).

$$$\lim_{x \to -\infty}\left(\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}\right)=-\infty$$$ (for steps, see limit calculator).

Thus, there are no horizontal asymptotes.

### Slant Asymptotes

Do polynomial long division $$$\frac{2 x^{3} + 15 x^{2} + 22 x - 11}{x^{2} + 8 x + 15}=2 x - 1 + \frac{4}{x^{2} + 8 x + 15}$$$ (for steps, see polynomial long division calculator).

The rational term approaches 0 as the variable approaches infinity.

Thus, the slant asymptote is $$$y=2 x - 1$$$.

**Answer**

**Vertical asymptotes:**$$$x=-5$$$; $$$x=-3$$$

**No horizontal asymptotes.**

**Slant asymptote:**$$$y=2 x - 1$$$

## Calculator vertical asymptote

.

Finding the asymptotes.

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