find all horizontal asymptotes of the following function.\n\nf(x) = \\frac{(3x - 8)(x - 4)}{2(3x - 8)}

Explanation:

Step1: Simplify the function

First, simplify (f(x)=\frac{(3x - 8)(x - 4)}{2(3x - 8)}). Since (3x-8\neq0) (when (x\neq\frac{8}{3})), we can cancel out the common factor (3x - 8). So (f(x)=\frac{x - 4}{2}=\frac{1}{2}x-2) for (x\neq\frac{8}{3}).

Step2: Analyze the limit as (x\to\pm\infty)

The limit of a linear function (y = mx + b) ((m\neq0)) as (x\to\pm\infty): (\lim_{x\to\infty}f(x)=\lim_{x\to\infty}(\frac{1}{2}x - 2)=\infty) (\lim_{x\to-\infty}f(x)=\lim_{x\to-\infty}(\frac{1}{2}x - 2)=-\infty)

Answer:

There are no horizontal asymptotes.