Question

find all horizontal asymptotes of the following function.

$f(x)=\frac{3(x + 4)(x - 4)}{2x(x + 4)}$

Explanation:

Step1: Simplify the function

First, simplify \(f(x)=\frac{3(x + 4)(x - 4)}{2x(x + 4)}\). Cancel out the common factor \((x + 4)\) (for \(x
eq - 4\)), so \(f(x)=\frac{3(x - 4)}{2x}=\frac{3x-12}{2x}=\frac{3}{2}-\frac{6}{x}\).

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

Use the limit formula \(\lim_{x
ightarrow\pm\infty}\frac{1}{x} = 0\).
For \(\lim_{x
ightarrow\infty}f(x)\), we have \(\lim_{x
ightarrow\infty}(\frac{3}{2}-\frac{6}{x})\).
Since \(\lim_{x
ightarrow\infty}\frac{6}{x}=0\), then \(\lim_{x
ightarrow\infty}(\frac{3}{2}-\frac{6}{x})=\frac{3}{2}\).
For \(\lim_{x
ightarrow-\infty}f(x)\), we have \(\lim_{x
ightarrow-\infty}(\frac{3}{2}-\frac{6}{x})\).
Since \(\lim_{x
ightarrow-\infty}\frac{6}{x}=0\), then \(\lim_{x
ightarrow-\infty}(\frac{3}{2}-\frac{6}{x})=\frac{3}{2}\).

Answer:

One Horizontal Asymptote \(y = \frac{3}{2}\)