역함수 합성 극한

$\displaystyle \lim\limits _{x\rightarrow 1} \frac{f ( x ) - 2}{x - 1} = \frac{1}{2}$에서 $f ( 1 ) = 2$, $\displaystyle f' ( 1 ) = \frac{1}{2}$이다. ∴ $g ( 2 ) = 1$, $\displaystyle g' ( 2 ) = \frac{1}{f' ( 1 )} = 2$ $\displaystyle \lim\limits _{x\rightarrow 2} \frac{f ( x ) - 3}{x - 2} = 4$에서 $f ( 2 ) = 3$, $f' ( 2 ) = 4$이다. ∴ $g ( 3 ) = 2$, $\displaystyle g' ( 3 ) = \frac{1}{f' ( 2 )} = \frac{1}{4}$ $h ( x ) = g ( g ( x ) )$라 하면 $h' ( x ) = g' ( g ( x ) ) g' ( x )$이고 $h ( 3 ) = g ( g ( 3 ) ) = g ( 2 ) = 1$이므로 $\displaystyle \lim\limits _{x\rightarrow 3} \frac{g ( g ( x ) ) - 1}{x - 3}$ $\displaystyle = \lim\limits _{x\rightarrow 3} \frac{h ( x ) - h ( 3 )}{x - 3}$ $= h' ( 3 )$ $= g' ( g ( 3 ) ) g' ( 3 )$ $= g' ( 2 ) g' ( 3 )$ $\displaystyle = 2 \times \frac{1}{4} = \frac{1}{2}$