한쪽 미분계수 극한

[출제의도] 지수함수의 미분을 이용하여 추론하기 조건 (나)에 의하여 $f \left( x + 2 k \right)$$\displaystyle = \mathit{-} \frac{1}{2} f \left( x + 2 \left( k - 1 \right) \right)$ $\displaystyle = \left( - \frac{1}{2} \right) ^{2} f \left( x + 2 \left( k - 2 \right) \right)$ $\vdots$ $\displaystyle = \left( - \frac{1}{2} \right) ^{k} f \left( x \right)$ ($k$는 자연수) 자연수 $m$에 대하여 $2 m - 2 \leq x \leq 2 m - 1$일 때 $f \left( x \right)$$= f \left( x - 2 \left( m - 1 \right) + 2 \left( m - 1 \right) \right)$ $\displaystyle = \left( - \frac{1}{2} \right) ^{m - 1} \times f \left( x - 2 \left( m - 1 \right) \right)$ $\displaystyle = \left( - \frac{1}{2} \right) ^{m - 1} \times \left\{ 2 ^{x - 2 \left( m - 1 \right)} - 1 \right\}$ $\displaystyle = \left( - \frac{1}{2} \right) ^{m - 1} \times \left\{ 2 ^{- 2 \left( m - 1 \right)} \times 2 ^{x} - 1 \right\}$ $2 m - 1 < x \leq 2 m$일 때 $f \left( x \right)$$= f \left( x - 2 \left( m - 1 \right) + 2 \left( m - 1 \right) \right)$ $\displaystyle = \left( - \frac{1}{2} \right) ^{m - 1} \times f \left( x - 2 \left( m - 1 \right) \right)$ $\displaystyle = \left( - \frac{1}{2} \right) ^{m - 1} \times \left\{ 4 \times \left( \frac{1}{2} \right) ^{x - 2 \left( m - 1 \right)} - 1 \right\}$ $\displaystyle = \left( - \frac{1}{2} \right) ^{m - 1} \times \left\{ 2 ^{2 m} \times \left( \frac{1}{2} \right) ^{x} - 1 \right\}$ 이므로 $2 m - 2 < x < 2 m - 1$에서 $\displaystyle f' \left( x \right) = \left( - \frac{1}{2} \right) ^{m - 1} \times 2 ^{- 2 \left( m - 1 \right)} \times 2 ^{x} \ln 2$ $2 m - 1 < x < 2 m$에서 $\displaystyle f' \left( x \right) = \mathit{-} \left( - \frac{1}{2} \right) ^{m - 1} \times 2 ^{2 m} \times \left( \frac{1}{2} \right) ^{x} \ln 2$ 자연수 $l$에 대하여 $2 l - 2 < x < 2 l - 1$ 또는 $2 l - 1 < x < 2 l$일 때 $g \left( x \right)$$\displaystyle = \lim\limits _{h \rightarrow 0 +} {\frac{f \left( x + h \right) - f \left( x \right) - \left\{ f \left( x - h \right) - f \left( x \right) \right\}}{h}}$$= 2 f' \left( x \right)$ $x = 2 l - 1$일 때 $g \left( x \right)$$\displaystyle = \lim\limits _{h \rightarrow 0 +} {\frac{f \left( 2 l - 1 + h \right) - f \left( 2 l - 1 - h \right)}{h}}$ $\displaystyle = \lim\limits _{h \rightarrow 0 +} {\left[ \frac{\left( - \frac{1}{2} \right) ^{l - 1} \left\{ 2 ^{2 l} \times \left( \frac{1}{2} \right) ^{2 l - 1 + h} - 1 \right\}}{h} \right.}$ $\displaystyle - \frac{\left( - \frac{1}{2} \right) ^{l - 1} \left\{ 2 ^{- 2 \left( l - 1 \right)} \times 2 ^{2 l - 1 - h} - 1 \right\}}{h} ]$ $\displaystyle = \left( - \frac{1}{2} \right) ^{l - 1} \times \lim\limits _{h \rightarrow 0 +} {\frac{\left( 2 ^{- h + 1} - 1 \right) - \left( 2 ^{- h + 1} - 1 \right)}{h}}$ $= 0$ $x = 2 l$일 때 $g \left( x \right)$$\displaystyle = \lim\limits _{h \rightarrow 0 +} {\frac{f \left( 2 l + h \right) - f \left( 2 l - h \right)}{h}}$ $\displaystyle = \lim\limits _{h \rightarrow 0 +} {\left[ \frac{\left( - \frac{1}{2} \right) ^{l} \left( 2 ^{- 2 l} \times 2 ^{2 l + h} - 1 \right)}{h} \right.}$$\displaystyle \left. - \frac{\left( - \frac{1}{2} \right) ^{l - 1} \left\{ 2 ^{2 l} \times \left( \frac{1}{2} \right) ^{2 l - h} - 1 \right\}}{h} \right]$ $\displaystyle = 3 \times \left( - \frac{1}{2} \right) ^{l} \times \lim\limits _{h \rightarrow 0 +} {\frac{2 ^{h} - 1}{h}}$ $\displaystyle = 3 \times \left( - \frac{1}{2} \right) ^{l} \ln 2$ 이제 $\displaystyle \lim\limits _{t \rightarrow 0 +} {\left\{ g \left( n + t \right) - g \left( n - t \right) \right\}} + 2 g \left( n \right) = \frac{\ln 2}{2 ^{24}}$를 만족시키는 자연수 $n$의 값을 $n$이 홀수일 때와 $n$이 짝수일 때로 나누어 구하면 다음과 같다. (ⅰ) $n = 2 s - 1$($s$는 자연수)일 때 $\displaystyle \lim\limits _{t \rightarrow 0 +} {g \left( n + t \right)}$ $\displaystyle = \lim\limits _{t \rightarrow 0 +} {g \left( 2 s - 1 + t \right)}$ $\displaystyle = \lim\limits _{t \rightarrow 0 +} {2 f' \left( 2 s - 1 + t \right)}$ $\displaystyle = \mathit{2} \times \left\{ - \left( - \frac{1}{2} \right) ^{s - 1} \right\} \times 2 ^{2 s} \times \left( \frac{1}{2} \right) ^{2 s - 1} \ln 2$ $\displaystyle = \mathit{8} \times \left( - \frac{1}{2} \right) ^{s} \ln 2$ $\displaystyle \lim\limits _{t \rightarrow 0 +} {g \left( n - t \right)}$ $\displaystyle = \lim\limits _{t \rightarrow 0 +} {g \left( 2 s - 1 - t \right)}$ $\displaystyle = \lim\limits _{t \rightarrow 0 +} {2 f' \left( 2 s - 1 - t \right)}$ $\displaystyle = 2 \times \mathit{\left( - \frac{1}{2} \right)} ^{s - 1} \times 2 ^{- 2 \left( s - 1 \right)} \times 2 ^{2 s - 1} \ln 2$ $\displaystyle = \mathit{-} 8 \times \left( - \frac{1}{2} \right) ^{s} \ln 2$ 그러므로 $\displaystyle \lim\limits _{t \rightarrow 0 +} {\left\{ g \left( n + t \right) - g \left( n - t \right) \right\}} + 2 g \left( n \right)$ $\displaystyle = 8 \times \left( - \frac{1}{2} \right) ^{s} \ln 2 - \left\{ - 8 \times \left( - \frac{1}{2} \right) ^{s} \ln 2 \right\} + 0$ $\displaystyle = 16 \times \left( - \frac{1}{2} \right) ^{s} \ln 2$ $\displaystyle 16 \times \left( - \frac{1}{2} \right) ^{s} \ln 2 = \frac{\ln 2}{2 ^{24}}$, $\displaystyle \left( - \frac{1}{2} \right) ^{s} = \left( \frac{1}{2} \right) ^{28}$ $s = 28$이므로 $n = 2 \times 28 - 1 = 55$ (ⅱ) $n = 2 s$($s$는 자연수)일 때 $\displaystyle \lim\limits _{t \rightarrow 0 +} {g \left( n + t \right)}$$\displaystyle = \lim\limits _{t \rightarrow 0 +} {g \left( 2 s + t \right)}$ $\displaystyle = \lim\limits _{t \rightarrow 0 +} {2 f' \left( 2 s + t \right)}$ $\displaystyle = 2 \times \left( - \frac{1}{2} \right) ^{s} \times 2 ^{- 2 s} \times 2 ^{2 s} \ln 2$ $\displaystyle = 2 \times \left( - \frac{1}{2} \right) ^{s} \ln 2$ $\displaystyle \lim\limits _{t \rightarrow 0 +} {g \left( n - t \right)}$$\displaystyle = \lim\limits _{t \rightarrow 0 +} {g \left( 2 s - t \right)}$ $\displaystyle = \lim\limits _{t \rightarrow 0 +} {2 f' \left( 2 s - t \right)}$ $\displaystyle = \mathit{2} \times \left\{ - \left( - \frac{1}{2} \right) ^{s - 1} \right\} \times 2 ^{2 s} \times \left( \frac{1}{2} \right) ^{2 s} \ln 2$ $\displaystyle = \mathit{4} \times \left( - \frac{1}{2} \right) ^{s} \ln 2$ 그러므로 $\displaystyle \lim\limits _{t \rightarrow 0 +} {\left\{ g \left( n + t \right) - g \left( n - t \right) \right\}} + 2 g \left( n \right)$ $=$$\displaystyle 2 \times \left( - \frac{1}{2} \right) ^{s} \ln 2 - 4 \times \left( - \frac{1}{2} \right) ^{s} \ln 2 + 6 \times \left( - \frac{1}{2} \right) ^{s} \ln 2$ $\displaystyle = 4 \times \left( - \frac{1}{2} \right) ^{s} \ln 2$ $\displaystyle 4 \times \left( - \frac{1}{2} \right) ^{s} \ln 2 = \frac{\ln 2}{2 ^{24}}$, $\displaystyle \left( - \frac{1}{2} \right) ^{s} = \left( \frac{1}{2} \right) ^{26}$ $s = 26$이므로 $n = 2 \times 26 = 52$ (ⅰ), (ⅱ)에 의하여 모든 자연수 $n$의 값의 합은 $55 + 52 = 107$