Question

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.
the horizontal asymptote is ( y=)
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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:

\(-8\)