Find each limit. Use $$ -\infty $$ or $$ \infty $$ when appropriate. $$ f(x)=-\frac{2 x-8}{(x-8)^{4}} $$ $$ \begin{array}{lll} ext { (A). } \lim _{x ightarrow 8^{-}} f(x) & ext { (B) } \lim _{x ightarrow 8^{+}} f(x) & ext { (C) } \lim _{x ightarrow 8} f(x)\end{array} $$ $$ \begin{array}{l} ext { A. Select the correct choice below and, if necessary, fill in the answer box to complete your choice } \ ext { A. } \lim _{x ightarrow 8^{-}} f(x)=\square ext { (Simplify your answer) } \ ext { B. The limit does not exist. }\end{array} $$

Question

Find each limit. Use $$ -\infty $$ or $$ \infty $$ when appropriate. $$ f(x)=-\frac{2 x-8}{(x-8)^{4}} $$ $$ \begin{array}{lll}	ext { (A). } \lim _{x ightarrow 8^{-}} f(x) & 	ext { (B) } \lim _{x ightarrow 8^{+}} f(x) & 	ext { (C) } \lim _{x ightarrow 8} f(x)\end{array} $$ $$ \begin{array}{l}	ext { A. Select the correct choice below and, if necessary, fill in the answer box to complete your choice } \ 	ext { A. } \lim _{x ightarrow 8^{-}} f(x)=\square 	ext { (Simplify your answer) } \ 	ext { B. The limit does not exist. }\end{array} $$

UpStudy AI · Accepted Answer

To evaluate the limits of the function $$ f(x) = -\frac{2x - 8}{(x - 8)^4} $$ as $$ x $$ approaches 8 from the left ($$ 8^- $$), from the right ($$ 8^+ $$), and overall ($$ 8 $$), we'll analyze the behavior of the numerator and the denominator separately.

### Simplifying the Function
First, let's rewrite the function for clarity:
$$
f(x) = -\frac{2x - 8}{(x - 8)^4} = -\frac{2(x - 4)}{(x - 8)^4}
$$

### Analyzing the Limits

1. **Limit as $$ x \rightarrow 8^- $$ (from the left):**
   - **Numerator:** $$ 2x - 8 $$ approaches $$ 2(8) - 8 = 8 $$.
   - **Denominator:** $$ (x - 8)^4 $$ approaches $$ 0^+ $$ (since any real number raised to an even power is positive).
   - **Overall Behavior:** The fraction becomes $$ -\frac{8}{0^+} $$, which means $$ f(x) $$ approaches $$ -\infty $$.

2. **Limit as $$ x \rightarrow 8^+ $$ (from the right):**
   - **Numerator:** $$ 2x - 8 $$ still approaches $$ 8 $$.
   - **Denominator:** $$ (x - 8)^4 $$ again approaches $$ 0^+ $$.
   - **Overall Behavior:** Similarly, the fraction becomes $$ -\frac{8}{0^+} $$, so $$ f(x) $$ approaches $$ -\infty $$.

3. **Limit as $$ x \rightarrow 8 $$ (overall):**
   - Since both one-sided limits approach $$ -\infty $$, the overall limit is also $$ -\infty $$.

### Final Answers

$$
\begin{aligned}
\text{(A)} \quad & \lim_{x \rightarrow 8^-} f(x) = -\infty \
\text{(B)} \quad & \lim_{x \rightarrow 8^+} f(x) = -\infty \
\text{(C)} \quad & \lim_{x \rightarrow 8} f(x) = -\infty \
\end{aligned}
$$
$$ \begin{aligned} \text{(A)} \quad & \lim_{x \rightarrow 8^-} f(x) = -\infty \ \text{(B)} \quad & \lim_{x \rightarrow 8^+} f(x) = -\infty \ \text{(C)} \quad & \lim_{x \rightarrow 8} f(x) = -\infty \ \end{aligned} $$

Find each limit. Use or when appropriate.

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