To solve this, we need to recognize that the sum resembles a Riemann sum for a definite integral of the form $\int_a^b f(x) \, dx$.

Part (a)

The general form of a Riemann sum for $\int_a^b f(x) \, dx$ is: $\lim_{n \to \infty} \sum_{i=1}^n f(x_i) \, \Delta x$ where:

• $\Delta x = \frac{b - a}{n}$,
• $x_i = a + i \, \Delta x$.

In our case, the expression inside the sum, $\frac{4i}{n^2(2 + \frac{4i}{n})}$, suggests:

1. $\Delta x$ is likely $\frac{1}{n}$, as it is the term dividing $f(x)$.
2. $x_i$ is represented by $\frac{4i}{n}$, because this term varies with $i$ and scales with $\frac{1}{n}$.

Given this, we deduce:

• $\Delta x = \frac{1}{n}$,
• $x_i = \frac{4i}{n}$.

Next, we identify $a$, $b$, and $f(x)$:

• Since $x_i = \frac{4i}{n}$ spans from $\frac{4 \cdot 1}{n} \approx 0$ to $\frac{4 \cdot n}{n} = 4$, we conclude $a = 0$ and $b = 4$.
• The function $f(x)$ must match the form inside the sum, with $x = \frac{4i}{n}$ substituted. Rewriting $\frac{4i}{n^2(2 + \frac{4i}{n})}$ in terms of $x = \frac{4i}{n}$, we get: $f(x) = \frac{x}{2 + x}.$

Thus, we have: [ \text{Answer to (a): } [\Delta x, x_i, a, b, f(x)] = \left[ \frac{1}{n}, \frac{4*i}{n}, 0, 4, \frac{x}{2 + x} \right]. ]

Part (b)

Now, we compute the value of the limit, which corresponds to the definite integral: $\int_0^4 \frac{x}{2 + x} \, dx.$

To evaluate this integral:

1. Substitute $u = 2 + x \Rightarrow du = dx$.
2. When $x = 0$, $u = 2$; and when $x = 4$, $u = 6$.
3. This transforms the integral to: $\int_2^6 \frac{u - 2}{u} \, du = \int_2^6 \left( 1 - \frac{2}{u} \right) du.$
4. Now, integrate term-by-term: $\int_2^6 1 \, du - \int_2^6 \frac{2}{u} \, du = \left[ u - 2 \ln |u| \right]_2^6.$
5. Plugging in the limits: $\left( 6 - 2 \ln 6 \right) - \left( 2 - 2 \ln 2 \right) = 4 - 2 \ln 3.$
6. Calculating $4 - 2 \ln 3$ to four decimal places: $4 - 2 \ln 3 \approx 1.8028.$

Thus, the answer for (b) is: $\text{Answer: } \lim_{n \to \infty} \sum_{i=1}^n \frac{4i}{n^2(2 + \frac{4i}{n})} = 1.8028.$

Let me know if you need more details or have any questions.

Here are some related questions to expand on this concept:

1. How can we identify the limits of integration in other Riemann sum problems?
2. What techniques can we use for more complicated integrals involving logarithms?
3. Why does the choice of $x_i$ matter in Riemann sums?
4. What is the significance of $\Delta x$ in terms of partitioning the interval?
5. How does substitution simplify complex integrals?

Tip: To recognize integrable functions in Riemann sums, match terms in the sum to an integral's format, then identify $\Delta x$ and $x_i$ accordingly.