Limit, Continuity and Differentiability: JEE Main 2023 Mathematics question

Let $a_{1}, a_{2}, a_{3}, \ldots, a_{\mathrm{n}}$ be $\mathrm{n}$ positive consecutive terms of an arithmetic progression. If $\mathrm{d} > 0$ is its common difference, then \lim_\limits{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots \ldots \ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}\right) is