무리식 수열의 극한

$\displaystyle \lim\limits _{n \rightarrow \infty} {} 2 n \left( \sqrt{n ^{2} + 4} - \sqrt{n ^{2} + 1} \right)$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \left\{ \begin{array}{l} \begin{matrix} \\ \end{matrix} 2 n \left( \sqrt{n ^{2} + 4} - \sqrt{n ^{2} + 1} \right) \end{array} \right.$ $\displaystyle \left. \times \frac{\sqrt{n ^{2} + 4} + \sqrt{n ^{2} + 1}}{\sqrt{n ^{2} + 4} + \sqrt{n ^{2} + 1}} \right\}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{2 n \left\{ \left( n ^{2} + 4 \right) - \left( n ^{2} + 1 \right) \right\}}{\sqrt{n ^{2} + 4} + \sqrt{n ^{2} + 1}}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{6 n}{\sqrt{n ^{2} + 4} + \sqrt{n ^{2} + 1}} = 3$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{6}{\sqrt{1 + \frac{4}{n ^{2}}} + \sqrt{1 + \frac{1}{n ^{2}}}} = 3$