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We are tasked with choosing \( k \) so that the given function is continuous on any interval. The function is defined as:

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

To make this function continuous, we need to ensure that the limit of \( f(x) \) as \( x \) approaches 3 from both sides equals \( f(3) = k \).

### Step 1: Simplify the expression for \( x \neq 3 \)
We begin by simplifying \( \frac{4x^3 - 12x^2}{x - 3} \).

Factor the numerator:

\[
4x^3 - 12x^2 = 4x^2(x - 3).
\]

Thus, the expression becomes:

\[
\frac{4x^2(x - 3)}{x - 3}.
\]

For \( x \neq 3 \), we can cancel \( x - 3 \) from the numerator and denominator, leaving:

\[
f(x) = 4x^2 \text{ for } x \neq 3.
\]

### Step 2: Find the limit as \( x \to 3 \)

Now, we compute the limit of \( f(x) = 4x^2 \) as \( x \to 3 \):

\[
\lim_{x \to 3} 4x^2 = 4(3)^2 = 4 \times 9 = 36.
\]

### Step 3: Set \( k = 36 \)

For the function to be continuous at \( x = 3 \), we must have:

\[
k = \lim_{x \to 3} f(x) = 36.
\]

Thus, the value of \( k \) that ensures continuity is \( k = 36 \).

---

Would you like further details, or do you have any questions about the steps?

Here are five related questions you can explore:

1. How is continuity of a function defined at a point?
2. What is the difference between continuity and differentiability?
3. How can you check if a piecewise function is continuous at a point?
4. What happens if the function has a discontinuity? How do you classify it?
5. How would you approach continuity for functions with multiple critical points?

**Tip:** Always check if a piecewise function is continuous at the boundaries by equating the limit of the function to the function's value at those points.

If possible, choose k so that the following function is continuous on any interval: f(x) = (4x^3 - 12x^2) / (x - 3) for x ≠ 3 and f(x) = k for x = 3.

Solve for k in a piecewise function to ensure continuity by simplifying and calculating limits. This problem focuses on using factorization and the definition of continuity for functions at specific points. Ideal for students studying pre-calculus or calculus concepts.

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