find all horizontal asymptotes of the following function.\nf(x)=\frac{6x^{2}+45x + 81}{2x^{2}+22x +…

Explanation:

Step1: Identify the degrees of numerator and denominator

The degree of the numerator $6x^{2}+45x + 81$ is $n = 2$, and the degree of the denominator $2x^{2}+22x + 36$ is $m=2$.

Step2: Use the horizontal - asymptote rule for equal - degree polynomials

When $n = m$, the horizontal asymptote is $y=\frac{a_{n}}{b_{m}}$, where $a_{n}$ is the leading coefficient of the numerator and $b_{m}$ is the leading coefficient of the denominator. Here, $a_{n}=6$ and $b_{m}=2$. So, $y=\frac{6}{2}=3$.

Answer:

$y = 3$