Ans: (A)
$$
\begin{aligned}
\text { Hint } &: \operatorname{lt}_{n \rightarrow \infty}\left(\frac{\sqrt{n+1}+\sqrt{n+2}+\sqrt{n+3}+\ldots \ldots+\sqrt{2 n-1}}{n^{3 / 2}}\right) \\
=& \operatorname{It}_{n \rightarrow \infty}\left(\sqrt{1+\frac{1}{n}}+\sqrt{1+\frac{2}{n}}+\ldots \ldots \ldots+\sqrt{1+\frac{n-1}{n}}\right) \frac{1}{n} \\
=& \operatorname{It}_{n \rightarrow \infty} \sum_{r=1}^{n-1} \frac{1}{n} \sqrt{1+\frac{r}{n}} \\
=& \int_{0}^{1} \sqrt{1+x} d x=\frac{2}{3} \cdot(2 \sqrt{2}-1)
\end{aligned}
$$
