로그·지수 수열 극한 합

ⅰ) 수열$\displaystyle \left\{ \frac{\log n}{\log 2 n} \right\}$의 극한은 $\displaystyle \begin{aligned} \alpha = \lim\limits _{n \rightarrow \infty} {} \frac{\log n}{\log 2 n} & = \lim\limits _{n \rightarrow \infty} {} \frac{\log n}{\log 2 + \log n} \end{aligned}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{1}{\frac{\log 2}{\log n} + 1} = \frac{1}{0 + 1} = 1$ ($\displaystyle \because \lim\limits _{n \rightarrow \infty} \log n = \infty$) ⅱ) 수열$\displaystyle \left\{ \left( 1 + \frac{2}{n} \right) ^{\frac{1}{n}} \right\}$의 극한은 $\displaystyle \beta = \lim\limits _{n \rightarrow \infty} \left( 1 + \frac{2}{n} \right) ^{\frac{1}{n}} = \left( 1 + 0 \right) ^{0} = 1$ $\therefore \alpha + \beta = 2$