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

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\neq - 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\rightarrow\pm\infty}\frac{1}{x} = 0). For (\lim_{x\rightarrow\infty}f(x)), we have (\lim_{x\rightarrow\infty}(\frac{3}{2}-\frac{6}{x})). Since (\lim_{x\rightarrow\infty}\frac{6}{x}=0), then (\lim_{x\rightarrow\infty}(\frac{3}{2}-\frac{6}{x})=\frac{3}{2}). For (\lim_{x\rightarrow-\infty}f(x)), we have (\lim_{x\rightarrow-\infty}(\frac{3}{2}-\frac{6}{x})). Since (\lim_{x\rightarrow-\infty}\frac{6}{x}=0), then (\lim_{x\rightarrow-\infty}(\frac{3}{2}-\frac{6}{x})=\frac{3}{2}).

Answer:

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