로그 극한 변형

$\displaystyle \lim\limits _{x \rightarrow \infty} {\left\{ f ( x ) \ln \left( 1 + \frac{1}{2 x} \right) \right\}} =$$\displaystyle \lim\limits _{x \rightarrow \infty} {\left\{ \frac{f ( x )}{2 x} \ln \left( 1 + \frac{1}{2 x} \right) ^{2x} \right\}}$$= 4$ 에서 $\displaystyle \lim\limits _{x\rightarrow \infty} \ln \left( 1 + \frac{1}{2 x} \right) ^{2x} = 1$이므로 $\displaystyle \lim\limits _{x\rightarrow \infty} \frac{f ( x )}{2 x} = 4$ 이다. $\therefore$ $\displaystyle \lim\limits _{x\rightarrow \infty} \frac{f ( x )}{x - 3} = \lim\limits _{x\rightarrow \infty} \frac{f ( x )}{2 x} \times \frac{2 x}{x - 3}$$= 8$