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To solve this problem, we need to determine whether the given function \( f(x) \) is continuous at \( a = 8 \).

### Step-by-Step Process:

1. **Check if \( f(8) \) is defined:**
   - From the piecewise function, we see that:
     \[
     f(x) = \begin{cases} 
     \frac{x^2 - 64}{x - 8}, & \text{if } x \neq 8 \\
     6, & \text{if } x = 8 
     \end{cases}
     \]
   - For \( x = 8 \), \( f(8) = 6 \). So, \( f(8) \) is defined.

2. **Check if \( \lim_{x \to 8} f(x) \) exists:**
   - For \( x \neq 8 \), we need to simplify the expression \( \frac{x^2 - 64}{x - 8} \):
     \[
     \frac{x^2 - 64}{x - 8} = \frac{(x - 8)(x + 8)}{x - 8} = x + 8 \quad \text{for } x \neq 8.
     \]
   - So, \( \lim_{x \to 8} f(x) = \lim_{x \to 8} (x + 8) = 8 + 8 = 16 \).

3. **Check if \( \lim_{x \to 8} f(x) = f(8) \):**
   - From the previous step, \( \lim_{x \to 8} f(x) = 16 \), but \( f(8) = 6 \). These values are not equal.

### Conclusion:
Since \( \lim_{x \to 8} f(x) \neq f(8) \), the function is **not continuous** at \( x = 8 \). This matches option **B**: "The function is not continuous at \( a = 8 \) because although \( f(8) = 6 \) is defined and \( \lim_{x \to 8} f(x) \) exists, \( f(8) \neq \lim_{x \to 8} f(x) \)."

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Would you like further details or have any questions?

Here are five related questions to expand your understanding:

1. What are the conditions for a function to be continuous at a point?
2. How would the function change if \( f(8) \) were defined as 16 instead of 6?
3. Can a function be continuous at a point if the limit does not exist?
4. How do we handle discontinuities in real-world applications of functions?
5. How does the concept of one-sided limits play into continuity at a point?

**Tip:** Simplifying piecewise functions to identify limits is key when dealing with continuity problems.

Determine whether the following function is continuous at a. Use the continuity checklist to justify your answer: 
\( f(x) = \begin{cases} \frac{x^2 - 64}{x - 8} & \text{if } x \neq 8 \\ 6 & \text{if } x = 8 \end{cases} \), \( a = 8 \).

Explore the conditions for the continuity of a piecewise function at a specific point. This problem involves determining the limits, checking the definition of the function at the point, and ensuring the limits match the function value. Suitable for Grade 12 and college-level calculus students.

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