content attribution\nquestion 3 · 1 point\nwhat is the horizontal asymptote of the graph of ( f(x)=\frac{9…

content attribution\nquestion 3 · 1 point\nwhat is the horizontal asymptote of the graph of ( f(x)=\frac{9 x^{2}+2}{x^{2}+x} ) ?\ngive your answer in the form ( y=a ).\nprovide your answer below:

content attribution\nquestion 3 · 1 point\nwhat is the horizontal asymptote of the graph of ( f(x)=\frac{9 x^{2}+2}{x^{2}+x} ) ?\ngive your answer in the form ( y=a ).\nprovide your answer below:

Answer

Explanation:

Step1: Divide numerator and denominator by (x^{2})

$$ \begin{align*} f(x)&=\frac{9x^{2}+2}{x^{2}+x}\ &=\frac{\frac{9x^{2}}{x^{2}}+\frac{2}{x^{2}}}{\frac{x^{2}}{x^{2}}+\frac{x}{x^{2}}}\ &=\frac{9 + \frac{2}{x^{2}}}{1+\frac{1}{x}} \end{align*} $$

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

As (x\to\pm\infty), (\frac{2}{x^{2}}\to0) and (\frac{1}{x}\to0) $$ \lim_{x\to\pm\infty}f(x)=\lim_{x\to\pm\infty}\frac{9 + \frac{2}{x^{2}}}{1+\frac{1}{x}}=\frac{9 + 0}{1+0}=9 $$

Answer:

(y = 9)