Answer
The function written in the fractional form as $f\left( x \right)=\frac{P\left( x \right)}{Q\left( x \right)}$ (where $P\left( x \right),Q\left( x \right)$ are the polynomial functions of x, and $Q\left( x \right)\ne 0$ ) is called the rational function. Then, the horizontal asymptotes can be found out as,
• If the degree of $P\left( x \right)$ is greater than the degree of $Q\left( x \right)$ , then there is no horizontal asymptote.
• If the degree of $P\left( x \right)$ is equal to the degree of $Q\left( x \right)$ , then the horizontal asymptote is given by the ratio of the coefficient of the highest degree term of x in the numerator and denominator.
• If the degree of $P\left( x \right)$ is less than the degree of $Q\left( x \right)$ , then $f\left( x \right)=0$ gives the horizontal asymptote.
Work Step by Step
A function $f\left( x \right)$ is a rational function if it is written in the form
$f\left( x \right)=\frac{P\left( x \right)}{Q\left( x \right)}$ …… (1)
Where $P\left( x \right),Q\left( x \right)$ are the polynomial functions of x in simplest form, and $Q\left( x \right)$ cannot be a zero function.
For Example, let $f\left( x \right)=\frac{5{{x}^{2}}}{3{{x}^{3}}+1}$
Compare it with equation (1),
Since, the degree of $P\left( x \right)$ is less than the degree of $Q\left( x \right)$ , then $f\left( x \right)=0$ gives the horizontal asymptote.
Let $f\left( x \right)=\frac{5{{x}^{3}}}{3{{x}^{3}}+1}$
Compare it with equation (1),
Since, the degree of $P\left( x \right)$ is equal to the degree of $Q\left( x \right)$ , then the horizontal asymptote is given by the ratio of the coefficient of the highest degree term of x in the numerator and denominator.
The horizontal asymptote is $\frac{5}{3}$.
Let $f\left( x \right)=\frac{5{{x}^{4}}}{3{{x}^{3}}+1}$
Compare it with equation (1),
Since, the degree of $P\left( x \right)$ is greater than the degree of $Q\left( x \right)$ , then there is no horizontal asymptote.
