무리식 수열 극한

[출제의도] 수열의 극한 계산하기 $\displaystyle \lim\limits _{n \rightarrow \infty} {\left( \sqrt{n ^{4} + 5 n ^{2} + 5} - n ^{2} \right)}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {\frac{\left( \sqrt{n ^{4} + 5 n ^{2} + 5} - n ^{2} \right) \left( \sqrt{n ^{4} + 5 n ^{2} + 5} + n ^{2} \right)}{\sqrt{n ^{4} + 5 n ^{2} + 5} + n ^{2}}}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {\frac{5 n ^{2} + 5}{\sqrt{n ^{4} + 5 n ^{2} + 5} + n ^{2}}}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {\frac{5 + \frac{5}{n ^{2}}}{\sqrt{1 + \frac{5}{n ^{2}} + \frac{5}{n ^{4}}} + 1}}$$\displaystyle = \frac{5}{2}$