This problem involves recognizing the expression as a Riemann sum and solving for the integral.

The given limit is:

$\lim_{n \to \infty} \sum_{j=1}^n \frac{6}{n} f\left(6 + \frac{3j}{n}\right) = 6$

Step-by-Step Explanation:

1. Recognize the Riemann Sum:
   The expression corresponds to the Riemann sum definition of a definite integral: $\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{j=1}^n \Delta x \cdot f(x_j),$ where:

   • $\Delta x = \frac{b-a}{n}$ (width of each subinterval),
   • $x_j = a + j \Delta x$ (sample point in the interval).

   Here:

   • $\Delta x = \frac{6}{n}$,
   • $x_j = 6 + \frac{3j}{n}$.

   From $\Delta x = \frac{6}{n}$, the interval length is $6$. This tells us $b - a = 6$.

   From $x_j = 6 + \frac{3j}{n}$, the interval starts at $x = 6$. Thus: $a = 6, \quad b = 6 + 3 = 9.$

   Therefore, the Riemann sum is approximating the integral: $\int_6^9 f(x) \, dx.$

2. Equating to the Given Value: The problem states that the Riemann sum converges to $6$. Thus: $\int_6^9 f(x) \, dx = 6.$

Final Answer:

$\boxed{6}$

Would you like a detailed explanation of any step? Let me know!

Related Questions:

1. What is a Riemann sum, and how does it relate to definite integrals?
2. How do you determine the bounds $a$ and $b$ of an integral from a Riemann sum?
3. How do you solve for $\Delta x$ and $x_j$ in a Riemann sum?
4. Can you explain how to evaluate definite integrals with substitution methods?
5. What are common mistakes when interpreting limits of Riemann sums?

Tip:

Always match the form of a Riemann sum to its integral by carefully identifying $\Delta x$, the sample points, and the interval bounds.