Question

find the limit of f(x) = (-2 + 9/x)/(4 - 9/x^2) as x approaches ∞ and as x approaches -∞. lim f(x) = -1/2 (type a simplified fraction.) x→∞ lim f(x) = (type a simplified fraction.) x→ -∞

Explanation:

Step1: Recall limit rules for $\frac{1}{x^n}$ as $x\to\pm\infty$

As $x\to\pm\infty$, $\lim_{x\to\pm\infty}\frac{1}{x}=0$ and $\lim_{x\to\pm\infty}\frac{1}{x^2}=0$.

Step2: Evaluate $\lim_{x\to -\infty}f(x)$

We have $f(x)=\frac{- 2+\frac{9}{x}}{4-\frac{9}{x^2}}$. Substitute $\lim_{x\to -\infty}\frac{1}{x}=0$ and $\lim_{x\to -\infty}\frac{1}{x^2}=0$ into the function.
\[

$$\begin{align*} \lim_{x\to -\infty}\frac{-2 + \frac{9}{x}}{4-\frac{9}{x^2}}&=\frac{-2+9\times\lim_{x\to -\infty}\frac{1}{x}}{4 - 9\times\lim_{x\to -\infty}\frac{1}{x^2}}\\ &=\frac{-2 + 9\times0}{4-9\times0}\\ &=-\frac{1}{2} \end{align*}$$

\]

Answer:

$-\frac{1}{2}$