\[\begin{split} &\lim_{n \to \infty} \frac{\sqrt{n}}{\sqrt{n+\sqrt{n+\sqrt{n}}}}=\\[16pt] &=\lim_{n \to \infty} \frac{\sqrt{n}}{\sqrt{n+\sqrt{n+\sqrt{n}}}}\ \frac{:\sqrt{n}}{:\sqrt{n}}=\\[16pt] &=\lim_{n \to \infty} \dfrac{\sqrt{\dfrac{n}{n}}}{\sqrt{\dfrac{n}{n}+\sqrt{\dfrac{n}{n^2}+\sqrt{\dfrac{n}{n^4}}}}}=\\[16pt] &=\lim_{n \to \infty} \dfrac{\sqrt{1}}{\sqrt{1+\sqrt{\dfrac{1}{n}+\sqrt{\dfrac{1}{n^3}}}}}=\\[16pt] &=\frac{1}{\sqrt{1}}=1 \end{split}\]