Find the limits in $$ a), b $$ ), and $$ c $$ ) below for the function $$ f(x)=\frac{4 x}{x-8} $$. Use $$ -\infty $$ and $$ \infty $$ when appropriate. a) Select the correct choice below and fill in any answer boxes in your choice. A. $$ \lim _{x \rightarrow 8^{-}} f(x)=\square $$ (Simplify your answer ) B. The limit does not exist and is neither $$ -\infty $$ nor $$ \infty $$. b) Select the correct choice below and fill in any answer boxes in your choice. A. lim $$ f(x)=\square $$ $$ x \rightarrow 8^{+} $$ (Simplify your answer.) B. The limit does not exist and is neither $$ -\infty $$ nor $$ \infty $$. Select the correct choice nelow and fill in any answer boxes in your choice. A. lim $$ f(x)=\square $$ $$ x \rightarrow 8 $$ (Simplify your answer.) B. The limit does not exist and is neither $$ -\infty $$ nor $$ \infty $$.

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Let's analyze the limits of the function $$ f(x) = \frac{4x}{x - 8} $$ as $$ x $$ approaches 8 from different directions.

### a) $$\lim_{x 	o 8^-} f(x)$$

**Approach from the left ($$ x 	o 8^- $$)**:
- As $$ x $$ approaches 8 from the left, $$ x - 8 $$ approaches 0 from the negative side.
- Thus, $$ \frac{4x}{x - 8} $$ becomes $$ \frac{32}{	ext{negative small number}} $$, which tends to $$ -\infty $$.

**Answer**:
**A.** $$ \lim_{x 	o 8^-} f(x) = -\infty $$

### b) $$\lim_{x 	o 8^+} f(x)$$

**Approach from the right ($$ x 	o 8^+ $$)**:
- As $$ x $$ approaches 8 from the right, $$ x - 8 $$ approaches 0 from the positive side.
- Thus, $$ \frac{4x}{x - 8} $$ becomes $$ \frac{32}{	ext{positive small number}} $$, which tends to $$ \infty $$.

**Answer**:
**A.** $$ \lim_{x 	o 8^+} f(x) = \infty $$

### c) $$\lim_{x 	o 8} f(x)$$

**Two-sided approach**:
- From part (a), as $$ x $$ approaches 8 from the left, $$ f(x) $$ tends to $$ -\infty $$.
- From part (b), as $$ x $$ approaches 8 from the right, $$ f(x) $$ tends to $$ \infty $$.
- Since the left-hand limit and right-hand limit are not equal (they approach different infinities), the two-sided limit does not exist.

**Answer**:
**B.** The limit does not exist and is neither $$ -\infty $$ nor $$ \infty $$.

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**Summary of Answers**:
- **a.** A. $$ \lim_{x 	o 8^-} f(x) = -\infty $$
- **b.** A. $$ \lim_{x 	o 8^+} f(x) = \infty $$
- **c.** B. The limit does not exist and is neither $$ -\infty $$ nor $$ \infty $$.
**a.** $$ \lim_{x 	o 8^-} f(x) = -\infty $$ **b.** $$ \lim_{x 	o 8^+} f(x) = \infty $$ **c.** The limit does not exist and is neither $$ -\infty $$ nor $$ \infty $$.

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