수열의 극한값 계산

$\displaystyle b _{n} = a _{n} \times \left( \sqrt{n ^{2} + 4} - n \right)$이라 하면 $\displaystyle \lim\limits _{n \rightarrow \infty} {} b _{n} = 6$이고 $\displaystyle a _{n} = \frac{b _{n}}{\sqrt{n ^{2} + 4} - n} = \frac{b _{n}}{4} \left( \sqrt{n ^{2} + 4} + n \right)$이므로 $\displaystyle \lim\limits _{n \rightarrow \infty} {} \frac{2 a _{n} + 6 n ^{2}}{na _{n} + 5}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{\frac{b _{n}}{2} \left( \sqrt{n ^{2} + 4} + n \right) + 6 n ^{2}}{\frac{nb _{n}}{4} \left( \sqrt{n ^{2} + 4} + n \right) + 5}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{\frac{b _{n}}{2} \times \frac{\sqrt{n ^{2} + 4} + n}{n ^{2}} + 6}{\frac{b _{n}}{4} \times \frac{\sqrt{n ^{2} + 4} + n}{n} + \frac{5}{n ^{2}}}$ $\displaystyle = \frac{\frac{6}{2} \times 0 + 6}{\frac{6}{4} \times 2 + 0} = \frac{6}{3} = 2$