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To find the asymptote(s) of the function \( f(x) = \frac{x^3 - 8}{2x^2} \), we need to consider both vertical and horizontal asymptotes.

### 1. **Vertical Asymptotes:**
Vertical asymptotes occur where the denominator of a function equals zero, provided the numerator is not zero at the same point.

The denominator is \( 2x^2 \). Setting it equal to zero:

\[
2x^2 = 0
\]

\[
x^2 = 0
\]

\[
x = 0
\]

Thus, there is a vertical asymptote at \( x = 0 \).

### 2. **Horizontal Asymptotes:**
Horizontal asymptotes describe the behavior of the function as \( x \) approaches infinity (\( x \to \infty \)) or negative infinity (\( x \to -\infty \)).

For rational functions, if the degree of the numerator is higher than that of the denominator, there is no horizontal asymptote. However, we can find the oblique (slant) asymptote by performing polynomial long division.

Given:
\[
f(x) = \frac{x^3 - 8}{2x^2}
\]

Let's perform the division:

- Divide \( x^3 \) by \( 2x^2 \) to get \( \frac{1}{2}x \).
- Multiply \( \frac{1}{2}x \) by \( 2x^2 \) to get \( x^3 \).
- Subtract \( x^3 \) from \( x^3 - 8 \) to get \( -8 \).

This gives:
\[
f(x) = \frac{1}{2}x - \frac{8}{2x^2}
\]

As \( x \) approaches infinity, the term \( \frac{8}{2x^2} \) approaches 0, so the function behaves like:
\[
f(x) \approx \frac{1}{2}x
\]

Thus, the slant (oblique) asymptote is \( y = \frac{1}{2}x \).

### Summary:
- **Vertical asymptote:** \( x = 0 \)
- **Slant asymptote:** \( y = \frac{1}{2}x \)

Would you like more details or have any questions?

Here are some related questions:
1. How do you find horizontal asymptotes for different types of functions?
2. Can you have both a horizontal and slant asymptote?
3. What is the significance of vertical asymptotes in real-world applications?
4. How would you perform polynomial long division for more complex functions?
5. What happens to the asymptotes if the function is modified slightly?

**Tip:** When analyzing asymptotes, always check the degrees of the numerator and denominator first to determine the type of asymptote.

Learn how to find vertical and slant asymptotes of a rational function with a detailed step-by-step solution. This problem covers key concepts in asymptotes, rational functions, and polynomial long division. Suitable for Grades 11-12.

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