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To ensure that the given piecewise function is continuous on any interval, the value of \( k \) must be chosen such that the function is continuous at \( x = 5 \). This means that the left-hand limit (as \( x \to 5 \)) of the first part of the function must equal the value of the function at \( x = 5 \), which is \( k \).

The function is given as:

\[
f(x) = 
\begin{cases} 
\frac{6x^5 - 30x^4}{x - 5} & x \neq 5 \\
k & x = 5.
\end{cases}
\]

### Steps:
1. First, simplify the expression \( \frac{6x^5 - 30x^4}{x - 5} \) for \( x \neq 5 \).
2. Then, compute the limit of this simplified expression as \( x \to 5 \).
3. Set that limit equal to \( k \) and solve for \( k \).

Let's simplify the expression first. 

The numerator is \( 6x^5 - 30x^4 \), which can be factored.

\[
6x^5 - 30x^4 = 6x^4(x - 5)
\]

Now, substituting back into the function:

\[
\frac{6x^5 - 30x^4}{x - 5} = \frac{6x^4(x - 5)}{x - 5}
\]

For \( x \neq 5 \), we can cancel the \( (x - 5) \) term:

\[
\frac{6x^4(x - 5)}{x - 5} = 6x^4
\]

Thus, the simplified form of the function for \( x \neq 5 \) is:

\[
f(x) = 6x^4 \quad \text{for} \quad x \neq 5.
\]

### Step 2: Calculate the limit as \( x \to 5 \).

\[
\lim_{x \to 5} 6x^4 = 6(5)^4 = 6 \times 625 = 3750.
\]

### Step 3: Set the limit equal to \( k \) for continuity at \( x = 5 \).

\[
k = 3750.
\]

Thus, to make the function continuous at \( x = 5 \), the value of \( k \) must be:

\[
k = 3750.
\]

### Would you like further details or clarifications?

Here are 5 related questions for practice:
1. What is the definition of a continuous function?
2. Why is factoring important when simplifying rational functions?
3. How do you determine the limit of a polynomial as \( x \to a \)?
4. What is the importance of left-hand and right-hand limits in continuity?
5. How would you handle discontinuities caused by other types of functions?

**Tip**: Always check the limit from both sides when evaluating continuity at a specific point.

If possible, choose k so that the following function is continuous on any interval: f(x) = (6x^5 - 30x^4)/(x - 5) when x ≠ 5, and k when x = 5.

Solve a piecewise function problem by determining the value of k for which the function is continuous at x = 5. This calculus-based problem involves simplifying polynomials, calculating limits, and applying the continuity definition. Suitable for high school students in advanced math courses.

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