find all horizontal asymptotes of the following function.\nf(x)=\frac{3x^{2}-10x - 8}{3x^{2}-36x + 96}\nno…

find all horizontal asymptotes of the following function.\nf(x)=\frac{3x^{2}-10x - 8}{3x^{2}-36x + 96}\nno horizontal asymptotes\none horizontal asymptote\ntwo horizontal asymptotes\none horizontal asymptote

find all horizontal asymptotes of the following function.\nf(x)=\frac{3x^{2}-10x - 8}{3x^{2}-36x + 96}\nno horizontal asymptotes\none horizontal asymptote\ntwo horizontal asymptotes\none horizontal asymptote

Answer

Explanation:

Step1: Identify degrees of numerator and denominator

The degree of the numerator $3x^{2}-10x - 8$ and the denominator $3x^{2}-36x + 96$ is $n = m=2$.

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

For a rational function $\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}$ with $n = m$, the horizontal - asymptote is $y=\frac{a_n}{b_m}$. Here $a_n = 3$ and $b_m = 3$. So $\lim_{x\to\pm\infty}\frac{3x^{2}-10x - 8}{3x^{2}-36x + 96}=\lim_{x\to\pm\infty}\frac{3-\frac{10}{x}-\frac{8}{x^{2}}}{3-\frac{36}{x}+\frac{96}{x^{2}}}= 1$.

Answer:

One Horizontal Asymptote