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

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: Evaluate the limit

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

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 $$ \begin{align*} \lim_{x\rightarrow-\infty}\frac{15-\frac{9}{x}}{5+\frac{9}{x}}&=\frac{15 - 0}{5+0}\ &=3 \end{align*} $$

Answer:

One Horizontal Asymptote (y = 3)