find all horizontal asymptotes of the following function.\nf(x)=\frac{2x^{2}+34x + 144}{9x^{2}-64}

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), $a_n = 2$ and $b_m=9$.

Step2: Calculate the horizontal - asymptote value

The horizontal asymptote is given by $y=\frac{2}{9}$ since for $f(x)=\frac{2x^{2}+34x + 144}{9x^{2}-64}$, as $x\to\pm\infty$, we consider the leading - terms of the numerator and denominator. The limit $\lim_{x\to\pm\infty}\frac{2x^{2}+34x + 144}{9x^{2}-64}=\lim_{x\to\pm\infty}\frac{2+\frac{34}{x}+\frac{144}{x^{2}}}{9-\frac{64}{x^{2}}}=\frac{2}{9}$.

Answer:

$y = \frac{2}{9}$