Question

find all horizontal asymptotes of the following function.
f(x)=\frac{x^{2}-16}{15x^{2}+81x + 84}

Explanation:

Step1: Recall the rule for horizontal asymptotes

For a rational function $f(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}$, if $n = m$, the horizontal - asymptote is $y=\frac{a_n}{b_m}$. Here, $n = 2$ (degree of the numerator) and $m = 2$ (degree of the denominator), $a_n = 1$ (coefficient of $x^2$ in the numerator) and $b_m=15$ (coefficient of $x^2$ in the denominator).

Step2: Calculate the value of the horizontal asymptote

The horizontal asymptote is given by $y=\frac{1}{15}$.

Answer:

$y = \frac{1}{15}$