Question

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question 3 · 1 point
what is the horizontal asymptote of the graph of ( f(x)=\frac{9 x^{2}+2}{x^{2}+x} ) ?
give your answer in the form ( y=a ).
provide your answer below:

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:

\(y = 9\)