find all horizontal asymptotes of the following function.\n\n f(x)=\frac{3(x - 8)(x + 6)}{2(2x - 7)(x + 6)}…

find all horizontal asymptotes of the following function.\n\n f(x)=\frac{3(x - 8)(x + 6)}{2(2x - 7)(x + 6)} \n\nanswer attempt 1 out of 2\n\n

find all horizontal asymptotes of the following function.\n\n f(x)=\frac{3(x - 8)(x + 6)}{2(2x - 7)(x + 6)} \n\nanswer attempt 1 out of 2\n\n

Answer

Explanation:

Step1: Simplify the function

Cancel out the common factor ((x + 6)) in the numerator and denominator. [ f(x)=\frac{3(x - 8)}{2(2x - 7)}=\frac{3x-24}{4x - 14} ]

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}), if (n = m), then (\lim_{x\to\pm\infty}y=\frac{a_n}{b_m}). Here (n = m=1), (a_1 = 3), (b_1=4). [ \lim_{x\to\pm\infty}\frac{3x-24}{4x - 14}=\lim_{x\to\pm\infty}\frac{3-\frac{24}{x}}{4-\frac{14}{x}}=\frac{3}{4} ]

Answer:

One Horizontal Asymptote