지수·로그 분수함수 보기

ㄱ. $1 < a < b$이고 $x > 1$이면 $1 < a ^{x} < b ^{x}$이고 $0 < \log _{b} x < \log _{a} x$이므로 $a ^{x} + \log _{b} x < b ^{x} + \log _{a} x$ $f ( x )$$\displaystyle = \frac{b ^{x} + \log _{a} x}{a ^{x} + \log _{b} x} > 1$ <참> ㄴ. $b < a < 1$이면 $\displaystyle \lim\limits _{x \rightarrow \infty} {a ^{x}} = \lim\limits _{x \rightarrow \infty} {b ^{x}} = 0$이고 $\displaystyle \lim\limits _{x \rightarrow \infty} {\begin{aligned} \log _{a} x = \lim\limits _{x \rightarrow \infty} {\log _{b} x} = - \infty \\ \end{aligned}}$이므로 $\displaystyle \lim\limits _{x \rightarrow \infty} {f ( x )} = \lim\limits _{x \rightarrow \infty} {\frac{b ^{x} + \log _{a} x}{a ^{x} + \log _{b} x}}$$\displaystyle = \lim\limits _{x \rightarrow \infty} {\frac{\frac{b ^{x}}{\log _{b} x} + \frac{\log _{a} x}{\log _{b} x}}{\frac{a ^{x}}{\log _{b} x} + 1}}$ $\displaystyle = \lim\limits _{x \rightarrow \infty} {\frac{\frac{b ^{x}}{\log _{b} x} + \frac{\frac{\log _{b} x}{\log _{b} a}}{\log _{b} x}}{\frac{a ^{x}}{\log _{b} x} + 1}}$$\displaystyle = \frac{1}{\log _{b} a} = \log _{a} b$ <거짓> ㄷ. $\displaystyle \lim\limits _{x \rightarrow 0 +} {a ^{x}} = \lim\limits _{x \rightarrow 0 +} {b ^{x}} = 1$이고 $x \rightarrow 0 +$일 때, $\log _{a} x$, $\log _{b} x$는 $\infty$ 또는 $- \infty$로 발산하므로 ㄴ과 같은 방법으로 계산하면 $\displaystyle \lim\limits _{x \rightarrow + 0} {f ( x )} = \lim\limits _{x \rightarrow + 0} {\frac{b ^{x} + \log _{a} x}{a ^{x} + \log _{b} x}}$$\displaystyle = \lim\limits _{x \rightarrow \infty} {\frac{\frac{b ^{x}}{\log _{b} x} + \frac{\frac{\log _{b} x}{\log _{b} a}}{\log _{b} x}}{\frac{a ^{x}}{\log _{b} x} + 1}}$$= \log _{a} b$ <참>