\(\text { Biết rằng } \lim \frac{n+\sqrt{n^{2}+1}}{\sqrt{n^{2}-n}-2}=a \sin \frac{\pi}{4}+b . \text { Tính } S=a^{3}+b^{3}\)

* Hướng dẫn giải

Ta có \(\lim \frac{n+\sqrt{n^{2}+1}}{\sqrt{n^{2}-n}-2}=\lim \frac{1+\sqrt{1+\frac{1}{n^{2}}}}{\sqrt{1-\frac{1}{n}}-\frac{2}{n}}=\frac{1+\sqrt{1}}{1}=2 \sqrt{2} \sin \frac{\pi}{4}\)

\(\longrightarrow\left\{\begin{array}{l} a=2 \sqrt{2} \\ b=0 \end{array} \longrightarrow S=8\right.\)