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Let's break down the given function \( f(x) \) and check if it's continuous at \( x = 6 \).

The function is defined as:

\[
f(x) = \begin{cases}
\frac{x^2 - 3x - 18}{x - 6} & \text{if } x \neq 6 \\
7 & \text{if } x = 6
\end{cases}
\]

### Step 1: Simplify the expression for \( f(x) \) when \( x \neq 6 \)

The numerator \( x^2 - 3x - 18 \) can be factored:

\[
x^2 - 3x - 18 = (x - 6)(x + 3)
\]

So,

\[
\frac{x^2 - 3x - 18}{x - 6} = \frac{(x - 6)(x + 3)}{x - 6} = x + 3 \quad \text{for} \quad x \neq 6
\]

Thus, the function becomes:

\[
f(x) = \begin{cases}
x + 3 & \text{if } x \neq 6 \\
7 & \text{if } x = 6
\end{cases}
\]

### Step 2: Check continuity at \( x = 6 \)

To be continuous at \( x = 6 \), the following conditions must be met:
1. \( \lim_{x \to 6} f(x) \) exists.
2. \( f(6) \) is defined.
3. \( \lim_{x \to 6} f(x) = f(6) \).

#### 1. Find \( \lim_{x \to 6} f(x) \)

Since \( f(x) = x + 3 \) for \( x \neq 6 \), we can compute the limit as \( x \to 6 \):

\[
\lim_{x \to 6} f(x) = \lim_{x \to 6} (x + 3) = 6 + 3 = 9
\]

#### 2. Check if \( f(6) \) is defined

From the piecewise definition, \( f(6) = 7 \).

#### 3. Compare \( \lim_{x \to 6} f(x) \) and \( f(6) \)

We found that:

\[
\lim_{x \to 6} f(x) = 9 \quad \text{and} \quad f(6) = 7
\]

Since \( \lim_{x \to 6} f(x) \neq f(6) \), the function is **not continuous at \( x = 6 \)**.

### Conclusion

The correct reasoning is option **b)**: The function is not continuous at \( x = 6 \) because \( \lim_{x \to 6} f(x) \neq f(6) \).

---

Would you like more details on any part of this explanation? Here are some related questions:

1. How do we check for limits when simplifying rational functions?
2. Why does the limit \( \lim_{x \to 6} f(x) \) not equal \( f(6) \)?
3. What does it mean for a function to be continuous at a point?
4. How would the function be redefined to make it continuous at \( x = 6 \)?
5. Can the limit still exist if the function is not continuous?

**Tip**: Always check both the limit and the function's value at the point to determine continuity!

Determine if f(x) is continuous at x = 6. If not, select the option with the correct reasoning as to why not.

Analyze the continuity of a piecewise function at x = 6 using limits and piecewise function analysis. Learn how to simplify the function and evaluate the limit to determine if the function is continuous. Suitable for high school math students.

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