The value of `("lim")_(nvecoo)sum_(r=1)^(4n)(sqrt(n))/(sqrt(r)(3sqrt(r)+sqrt(n))^2)` is equal to `1/(35)` (b) `1/4` (c) `1/(10)` (d) `1/5`

Options

A. `1/35`
B. `1/14`
C. `1/10`
D. `1/5`

Explanation

To solve the limit problem, we will go through the steps systematically.

### Step 1: Write the limit expression
We start with the limit expression given in the problem:
\[ \lim_{n \to \infty} \sum_{r=1}^{4n} \frac{\sqrt{n}}{\sqrt{r} (3\sqrt{r} + \sqrt{n})^2} \]

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