\[ \begin{split} &\lim_{n \to \infty} \left(\sqrt{n+6\sqrt{n}+1}-\sqrt{n}\right)=\\[16pt] &=\lim_{n \to \infty} \left(\sqrt{n+6\sqrt{n}+1}-\sqrt{n}\right)\cdot \frac{\sqrt{n+6\sqrt{n}+1}+\sqrt{n}}{\sqrt{n+6\sqrt{n}+1}+\sqrt{n}}=\\[16pt] &=\lim_{n \to \infty} \frac{n+6\sqrt{n}+1-n}{\sqrt{n+6\sqrt{n}+1}+\sqrt{n}}=\\[16pt] &=\lim_{n \to \infty} \frac{6\sqrt{n}+1}{\sqrt{n+6\sqrt{n}+1}+\sqrt{n}}\frac{:\sqrt{n}}{:\sqrt{n}}=\\[16pt] &=\lim_{n \to \infty} \dfrac{6\sqrt{\dfrac{n}{n}}+\sqrt{\dfrac{1}{n}}}{\sqrt{\dfrac{n}{n}+6\sqrt{\dfrac{n}{n^2}}+\dfrac{1}{n}}+\sqrt{\dfrac{n}{n}}}=\\[16pt] &=\lim_{n \to \infty} \dfrac{6+\sqrt{\dfrac{1}{n}}}{\sqrt{1+6\sqrt{\dfrac{1}{n}}+\dfrac{1}{n}}+1}=\\[16pt] &= \frac{6+\sqrt{0}}{\sqrt{1+0+0}+1}=\frac{6}{2}=3 \end{split} \]