\[ \begin{split} &\lim_{n \to \infty} \frac{{\sqrt{n^2+\sqrt{n+1}}}-\sqrt{n^2-\sqrt{n-1}}}{\sqrt{n+1}-\sqrt{n}}=\\[16pt] &=\lim_{n \to \infty} \frac{{\sqrt{n^2+\sqrt{n+1}}}-\sqrt{n^2-\sqrt{n-1}}}{\sqrt{n+1}-\sqrt{n}}\cdot \frac{({\sqrt{n^2+\sqrt{n+1}}}+\sqrt{n^2-\sqrt{n-1}})(\sqrt{n+1}+\sqrt{n})}{({\sqrt{n^2+\sqrt{n+1}}}+\sqrt{n^2-\sqrt{n-1}})(\sqrt{n+1}+\sqrt{n})} =\\[16pt] &=\lim_{n \to \infty} \frac{(n^2+\sqrt{n+1}-n^2+\sqrt{n-1})(\sqrt{n+1}+\sqrt{n})}{\left(\sqrt{n^2+\sqrt{n+1}}+\sqrt{n^2-\sqrt{n-1}}\right)(n+1-n)}=\\[16pt] &=\lim_{n \to \infty} \frac{(\sqrt{n+1}+\sqrt{n-1})(\sqrt{n+1}+\sqrt{n})}{\left(\sqrt{n^2+\sqrt{n+1}}+\sqrt{n^2-\sqrt{n-1}}\right)}=\\[16pt] &=\lim_{n \to \infty} \frac{(n+1+\sqrt{n^2+n}+\sqrt{n^2-1}+\sqrt{n^2-n})}{\left(\sqrt{n^2+\sqrt{n+1}}+\sqrt{n^2-\sqrt{n-1}}\right)}\frac{:n}{:n}=\\[16pt] &=\lim_{n \to \infty} \frac{1+\frac{1}{n}+\sqrt{1+\frac{1}{n}}+\sqrt{1-\frac{1}{n^2}}+\sqrt{1-\frac{1}{n}}}{\sqrt{1+\sqrt{\frac{1}{n^3}+\frac{1}{n^4}}}+\sqrt{1-\sqrt{\frac{1}{n^3}-\frac{1}{n^4}}}}=\\[16pt] &=\frac{1+0+1+1+1}{\sqrt{1}+\sqrt{1}}=\frac{4}{2}=2 \end{split} \]