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How to find horizontal Asymptotes 

How to find horizontal Asymptotes?
A horizontal asymptotes is a horizontal line that tells us how the function will behave at every edges of the graph.
A function is an equation how two things are related with each other.In general the function tells us how Y is related to X.Usually the functions are often graphed for visualization. The horizontal asymptotes are parallel to X-axis some times it crosses or cuts the graph.
Horizontal asymptotes exists when the numerator and denominator of the function is a polynomials. So we called these functions as rational expressions.

Steps for how to find Horizontal Asymptotes

1) Write the given equation in y = form.
2) If there are factors given in the numerator and denominator then multiply them and write it in the form of polynomial.
3) Check the degree of numerator and denominator.
4) If the degrees are same then horizontal asymptote $y=\frac{(numerator's leading coefficient)}{ (denominator's leading coefficient)}$

5) If the degree of the denominator greater than the degree of numerator then the horizontal asymptote y = 0 which is nothing but the X-axis.

Examples : 1) Find the horizontal asymptote of the rational function $f(x)=\frac{3x^2-5x}{x^2-5x + 6} $

Solution: First we will write the given function in y form.
$y = f(x)=\frac{3x^2-5x}{x^2-5x + 6} $

Now we will check the degree of numerator and denominator.
Degree of numerator = 2 and the degree of the denominator =2 . Both the degrees are same so horizontal asymptote is given by the formula
$y=\frac{(numerator's leading coefficient)}{ (denominator's leading coefficient)}$

Numerator's leading coefficient = 3 and denominator's leading coefficient = 1
So the horizontal asymptote is $y =\frac{3}{ 1}$

2) $f(x)=\frac{x^3-5}{x-6} $

Solution : First we will write the given function in y form.
$y= f(x)=\frac{x^3-5}{x-6} $
Now we will check the degree of numerator and denominator.
Degree of numerator = 3 and the degree of the denominator =1 . Both the degrees are different so there is no horizontal asymptote. There is a slant asymptote, that you will learn in the next topic.

3) $f(x)=\frac{x-8}{x^3-1}$

Solution : First we will write the given function in y form.
$y= f(x)=\frac{x -8}{x^3-1} $
Now we will check the degree of numerator and denominator.
Degree of numerator = 1 and the degree of the denominator = 3
Since the degree of numerator is less than the degree of denominator.
So the horizontal asymptote is y = 0.

4) Find the horizontal asymptote from the given graph.
Rational function
Solution : From the graph, we observe that the horizontal asymptote is at y = 2.
Horizontal asymptote

Practice Questions

Q.1 Find the horizontal asymptote of the following function.
1) $f(x)=\frac{4x^3}{x^2-4x+2}$

2) $f(x)=\frac{7x + 2}{x+2}$

3) $f(x)=\frac{4x + 2}{x^2 - 5}$

11th grade math

Precalculus

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