로그 극한

[출제의도] 이해능력-지수함수와 로그함수 $\displaystyle \lim\limits _{x \rightarrow \infty} {} x \left\{ \ln ( 2 x + 5 ) - \ln ( 2 x - 1 ) \right\}$ $\displaystyle = \lim\limits _{x \rightarrow \infty} {} \left( x \ln \frac{2 x + 5}{2 x - 1} \right) = \lim\limits _{x \rightarrow \infty} {} \left\{ x \ln \left( 1 + \frac{6}{2 x - 1} \right) \right\}$ $\displaystyle = \lim\limits _{x \rightarrow \infty} {} \ln \left( 1 + \frac{6}{2 x - 1} \right) ^{x}$ $\displaystyle = \lim\limits _{x \rightarrow \infty} {} \ln \left\{ \left( 1 + \frac{6}{2 x - 1} \right) ^{\frac{2 x - 1}{6}} \right\} ^{\frac{6 x}{2 x - 1}}$ $\displaystyle = \lim\limits _{x \rightarrow \infty} {} \left\{ \frac{6 x}{2 x - 1} \times \ln \left( 1 + \frac{6}{2 x - 1} \right) ^{\frac{2 x - 1}{6}} \right\}$ $\displaystyle = \lim\limits _{x \rightarrow \infty} {} \frac{6 x}{2 x - 1} \times \lim\limits _{x \rightarrow \infty} {} \ln \left( 1 + \frac{6}{2 x - 1} \right) ^{\frac{2 x - 1}{6}}$ $= 3 \times \ln e = 3$