ln 극한

$f ( x ) = \ln x$라 할 때, $\displaystyle f' ( x ) = \frac{1}{x}$이므로 $\displaystyle \lim\limits _{x\rightarrow 1} \frac{\ln x}{x ^{3} - 1}$ $\displaystyle = \lim\limits _{x\rightarrow 1} \frac{\ln x}{x - 1} \cdot \frac{1}{x ^{2} + x + 1}$ $\displaystyle = \lim\limits _{x\rightarrow 1} \frac{f ( x ) - f ( 1 )}{x - 1} \cdot \frac{1}{x ^{2} + x + 1}$ $\displaystyle = f' ( 1 ) \cdot \frac{1}{3}$ $\displaystyle = \frac{1}{3}$ [다른 풀이] $\displaystyle \lim\limits _{x\rightarrow 1} \frac{\ln x}{x ^{3} - 1}$ $\displaystyle = \lim\limits _{x\rightarrow 1} \frac{\ln x}{x - 1} \cdot \frac{1}{x ^{2} + x + 1}$ $\displaystyle = \lim\limits _{t\rightarrow 0} \frac{\ln ( 1 + t )}{t} \cdot \lim\limits _{x\rightarrow 1} \frac{1}{x ^{2} + x + 1}$ $\displaystyle = 1 \cdot \frac{1}{3}$$\displaystyle = \frac{1}{3}$