find the horizontal asymptote of ( f(x)=6\frac{(x + 6)(8x - 1)}{(3 - x)(6x + 2)} ). if the horizontal…

find the horizontal asymptote of ( f(x)=6\frac{(x + 6)(8x - 1)}{(3 - x)(6x + 2)} ). if the horizontal asymptote does not exist, enter dne.\nthe horizontal asymptote is ( y=) \nquestion help: message instructor post to forum

find the horizontal asymptote of ( f(x)=6\frac{(x + 6)(8x - 1)}{(3 - x)(6x + 2)} ). if the horizontal asymptote does not exist, enter dne.\nthe horizontal asymptote is ( y=) \nquestion help: message instructor post to forum

Answer

Explanation:

Step1: Expand the numerator and denominator

  • Expand ((x + 6)(8x-1)=8x^{2}-x + 48x-6=8x^{2}+47x - 6)
  • Expand ((3 - x)(6x + 2)=18x+6-6x^{2}-2x=-6x^{2}+16x + 6)
  • So (f(x)=6\frac{8x^{2}+47x - 6}{-6x^{2}+16x + 6})

Step2: Use the rule for horizontal asymptotes of rational functions

For a rational function (y = a\frac{f(x)}{g(x)}) where (f(x)=a_nx^n+\cdots) and (g(x)=b_mx^m+\cdots) If (n = m), the horizontal asymptote is (y=a\frac{a_n}{b_m}) Here (n = m = 2), (a = 6), (a_n=8), (b_m=-6)

Step3: Calculate the value of the horizontal asymptote

(y = 6\times\frac{8}{-6}=-8)

Answer:

(-8)