question\nfind all horizontal asymptotes of the following function.\n\nf(x)=\\frac{15x - 9}{5x + 9}

Explanation:

Step1: Divide numerator and denominator by (x)

$$ \begin{align*} \lim_{x\rightarrow\infty}\frac{15x - 9}{5x+9}&=\lim_{x\rightarrow\infty}\frac{\frac{15x}{x}-\frac{9}{x}}{\frac{5x}{x}+\frac{9}{x}}\ &=\lim_{x\rightarrow\infty}\frac{15-\frac{9}{x}}{5 + \frac{9}{x}} \end{align*} $$

Step2: Use the limit property (\lim_{x\rightarrow\infty}\frac{c}{x}=0) ((c) is a constant)

As (x\rightarrow\infty), (\lim_{x\rightarrow\infty}\frac{9}{x}=0). Then (\lim_{x\rightarrow\infty}\frac{15-\frac{9}{x}}{5+\frac{9}{x}}=\frac{15 - 0}{5+0}=3)

Step3: Check the limit as (x\rightarrow-\infty)

$$ \begin{align*} \lim_{x\rightarrow-\infty}\frac{15x - 9}{5x+9}&=\lim_{x\rightarrow-\infty}\frac{\frac{15x}{x}-\frac{9}{x}}{\frac{5x}{x}+\frac{9}{x}}\ &=\lim_{x\rightarrow-\infty}\frac{15-\frac{9}{x}}{5+\frac{9}{x}} \end{align*} $$ As (x\rightarrow-\infty), (\lim_{x\rightarrow-\infty}\frac{9}{x}=0). So (\lim_{x\rightarrow-\infty}\frac{15-\frac{9}{x}}{5+\frac{9}{x}}=\frac{15-0}{5 + 0}=3)

Answer:

The horizontal asymptote is (y = 3)