find all horizontal asymptotes of the following function. f(x)=(3x - 7)(x + 6)/2(x + 9)(x + 6) answer one…

Answer:

$y = \frac{3}{2}$

Explanation:

Step1: Simplify the function

First, cancel out the common factor $(x + 6)$ in the numerator and denominator. So $f(x)=\frac{3x - 7}{2(x + 9)}=\frac{3x-7}{2x + 18}$.

Step2: Find the limit as $x\to\pm\infty$

For a rational - function $y=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}$, when $n = m$, the horizontal asymptote is $y=\frac{a_n}{b_m}$. Here, $n = m = 1$, $a_1 = 3$ and $b_1 = 2$. Calculate $\lim_{x\to\pm\infty}\frac{3x-7}{2x + 18}$. Divide both the numerator and denominator by $x$: $\lim_{x\to\pm\infty}\frac{3-\frac{7}{x}}{2+\frac{18}{x}}$. As $x\to\pm\infty$, $\frac{7}{x}\to0$ and $\frac{18}{x}\to0$. So $\lim_{x\to\pm\infty}\frac{3-\frac{7}{x}}{2+\frac{18}{x}}=\frac{3}{2}$.