Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ f(x)=\frac{8 x}{x^{2}-4} $$ Find any horizontal asymptotes of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, $$ \square $$ (Type an equation.) B. The function has two horzontal asymptotes. The top asymptote is $$ \square $$ and the bottom asymptote is $$ \square $$ . (Type equations.) C. There are no horizontal asymptotes. Find any vertical asymptotes of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, $$ \square $$ . (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is $$ \square $$ and the rightmost asymptote is $$ \square $$ . (Type equations.) C. There are no vertical asymplotes.

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To analyze the function $$ f(x) = \frac{8x}{x^2 - 4} $$, we will follow these steps:

### Step 1: Identify Horizontal Asymptotes

Horizontal asymptotes are determined by examining the behavior of the function as $$ x $$ approaches infinity or negative infinity.

1. **Degree of the numerator and denominator**:
   - The degree of the numerator $$ 8x $$ is 1.
   - The degree of the denominator $$ x^2 - 4 $$ is 2.

2. **Horizontal asymptote rule**:
   - If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $$ y = 0 $$.
   - If the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients.

Since the degree of the numerator (1) is less than the degree of the denominator (2), we conclude that:

$$
\text{Horizontal Asymptote: } y = 0
$$

### Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points).

1. **Set the denominator to zero**:
   $$
   x^2 - 4 = 0
   $$
   $$
   x^2 = 4
   $$
   $$
   x = 2 \quad \text{or} \quad x = -2
   $$

Thus, the vertical asymptotes are at $$ x = 2 $$ and $$ x = -2 $$.

### Summary of Asymptotes

- **Horizontal Asymptote**: $$ y = 0 $$
- **Vertical Asymptotes**: $$ x = -2 $$ and $$ x = 2 $$

### Step 3: Sketch the Graph

To sketch the graph of $$ f(x) = \frac{8x}{x^2 - 4} $$:

1. **Plot the horizontal asymptote** at $$ y = 0 $$.
2. **Plot the vertical asymptotes** at $$ x = -2 $$ and $$ x = 2 $$.
3. **Determine the behavior of the function** in the intervals defined by the vertical asymptotes:
   - For $$ x < -2 $$
   - For $$ -2 < x < 2 $$
   - For $$ x > 2 $$

### Step 4: Analyze the Function Behavior

- As $$ x \to -\infty $$, $$ f(x) \to 0 $$ (approaching the horizontal asymptote).
- As $$ x \to -2^- $$, $$ f(x) \to +\infty $$.
- As $$ x \to -2^+ $$, $$ f(x) \to -\infty $$.
- As $$ x \to 2^- $$, $$ f(x) \to +\infty $$.
- As $$ x \to 2^+ $$, $$ f(x) \to -\infty $$.
- As $$ x \to +\infty $$, $$ f(x) \to 0 $$ (approaching the horizontal asymptote).

### Final Answers

1. **Horizontal Asymptote**: 
   - A. The function has one horizontal asymptote, $$ y = 0 $$.

2. **Vertical Asymptotes**: 
   - B. The function has two vertical asymptotes. The leftmost asymptote is $$ x = -2 $$ and the rightmost asymptote is $$ x = 2 $$.

### Graph Sketch

The graph will show the function approaching the horizontal asymptote $$ y = 0 $$ as $$ x $$ moves towards $$ \pm \infty $$ and will have vertical asymptotes at $$ x = -2 $$ and $$ x = 2 $$, with the function diverging to $$ +\infty $$ and $$ -\infty $$ around these points.
The function $$ f(x) = \frac{8x}{x^2 - 4} $$ has: - **Horizontal Asymptote**: $$ y = 0 $$ - **Vertical Asymptotes**: $$ x = -2 $$ and $$ x = 2 $$ The graph approaches $$ y = 0 $$ as $$ x $$ approaches $$ \pm \infty $$ and has vertical asymptotes at $$ x = -2 $$ and $$ x = 2 $$.

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