Question

find all horizontal asymptotes of the following function.

f(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 \(3x-8
eq0\) (when \(x
eq\frac{8}{3}\)), we can cancel out the common factor \(3x - 8\). So \(f(x)=\frac{x - 4}{2}=\frac{1}{2}x-2\) for \(x
eq\frac{8}{3}\).

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

The limit of a linear function \(y = mx + b\) (\(m
eq0\)) 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.