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To solve the given problem, we need to make the function \( f(x) \) continuous at \( x = 4 \). The function \( f(x) \) is defined as:

\[
f(x) = \begin{cases} 
\frac{x^2 - x - 12}{x^2 - 3x - 4} & \text{for } x \neq 4 \\
k & \text{for } x = 4 
\end{cases}
\]

### Step 1: Simplify the function for \( x \neq 4 \)
First, simplify the expression \( \frac{x^2 - x - 12}{x^2 - 3x - 4} \).

- Factor the numerator:
  \[
  x^2 - x - 12 = (x - 4)(x + 3)
  \]

- Factor the denominator:
  \[
  x^2 - 3x - 4 = (x - 4)(x + 1)
  \]

So, the function becomes:
\[
f(x) = \frac{(x - 4)(x + 3)}{(x - 4)(x + 1)} \quad \text{for} \quad x \neq 4
\]

Cancel out \( (x - 4) \) since it is present in both the numerator and denominator:
\[
f(x) = \frac{x + 3}{x + 1} \quad \text{for} \quad x \neq 4
\]

### Step 2: Ensure continuity at \( x = 4 \)
To make the function continuous at \( x = 4 \), the following condition must hold:
\[
\lim_{x \to 4} f(x) = f(4)
\]

Now, compute the limit of \( f(x) \) as \( x \) approaches 4:
\[
\lim_{x \to 4} f(x) = \lim_{x \to 4} \frac{x + 3}{x + 1} = \frac{4 + 3}{4 + 1} = \frac{7}{5}
\]

### Step 3: Set \( k = \lim_{x \to 4} f(x) \)
For continuity, \( f(4) = k \), so we must have:
\[
k = \frac{7}{5}
\]

Thus, the value of \( k \) that makes the function continuous at \( x = 4 \) is:
\[
k = \frac{7}{5}
\]

Would you like more details or have any questions? Here are some related questions to further explore the topic:

1. What does it mean for a function to be continuous at a point?
2. How do you handle cases where the numerator and denominator cancel out in rational functions?
3. What are the conditions for a piecewise function to be continuous at a specific point?
4. How can limits be used to determine the continuity of a function?
5. What are some real-life applications of continuous functions in calculus?

**Tip**: Always check if you can factor and simplify rational functions, as this can help resolve potential discontinuities in the function.

Find all values of k that would make the function f(x) continuous at x = 4. f(x) = (x^2 - x - 12) / (x^2 - 3x - 4) for x ≠ 4 and k for x = 4.

Learn how to find the value of k that makes the function f(x) continuous at x = 4. This problem involves simplifying rational functions, calculating limits, and applying the continuity condition. Detailed step-by-step solution provided. Suitable for Grade 11-12 or Calculus students.

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