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To find the vertical asymptotes of the given rational function:

\[
f(x) = \frac{x^2 - 11x + 24}{x^{12} - 8x^{11}}
\]

We focus on the **denominator** of the function, since vertical asymptotes occur where the denominator is equal to zero (but the numerator is non-zero).

### Steps to find vertical asymptotes:
1. **Denominator**: \( x^{12} - 8x^{11} \)
   
   Factor the denominator:
   \[
   x^{11}(x - 8) = 0
   \]
   
   So, the denominator equals zero when:
   \[
   x^{11} = 0 \quad \text{or} \quad x - 8 = 0
   \]
   This gives two solutions:
   \[
   x = 0 \quad \text{or} \quad x = 8
   \]

2. **Check the numerator**: \( x^2 - 11x + 24 \)
   
   We need to ensure that the numerator is not zero at these points. Let’s substitute the values of \( x = 0 \) and \( x = 8 \) into the numerator:
   
   - For \( x = 0 \):
     \[
     f(0) = 0^2 - 11(0) + 24 = 24 \quad (\text{non-zero})
     \]
   - For \( x = 8 \):
     \[
     f(8) = 8^2 - 11(8) + 24 = 64 - 88 + 24 = 0
     \]
   
   Since the numerator equals zero at \( x = 8 \), **no vertical asymptote** occurs at this point because there is a common factor in both the numerator and the denominator.

### Conclusion:
The only vertical asymptote occurs at \( x = 0 \).

Thus, the correct choice is **A**, with \( x = 0 \) as the vertical asymptote.

Would you like further details or have any questions?

Here are some related questions for further practice:
1. How do you identify horizontal asymptotes for rational functions?
2. What happens at points where both the numerator and denominator are zero?
3. How do you find the slant asymptotes of a rational function?
4. What are the conditions for a rational function to have no vertical asymptotes?
5. How can graphing help to visually confirm the location of vertical asymptotes?

**Tip:** When both the numerator and denominator are zero at the same point, it typically indicates a hole (removable discontinuity) rather than an asymptote.

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following function: f(x) = (x^2 - 11x + 24) / (x^12 - 8x^11).

This problem involves identifying vertical asymptotes for a rational function by factoring the denominator and checking if the numerator equals zero at specific points. Detailed steps are provided for solving this problem using algebraic methods.

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