선분 비의 극한

[출제의도] 선분의 길이를 이용하여 지수함수의 극한값을 구한다. $\mathrm{A} ( \mathit{t} , 2 ^{t} )$, $\displaystyle \left. \mathrm{B} \left( \mathit{t} , \left( \frac{1}{2} \right) ^{t} \right) \right.$, $\mathrm{H} ( \mathit{0} , 2 ^{t} )$에서 $\displaystyle \overline{\mathrm{A}} H = \mathit{t}$이고, $\displaystyle \mathrm{\overline{AB}} = \mathit{2} ^{t} - \left( \frac{1}{2} \right) ^{t}$이므로 $\displaystyle \lim\limits _{t \rightarrow 0 +} \frac{\mathrm{\overline{AB}}}{\mathrm{\overline{AH}}} = \lim\limits _{t \rightarrow 0 +} \frac{2 ^{t} - \left( \frac{1}{2} \right) ^{t}}{t}$ $\displaystyle = \lim\limits _{t \rightarrow 0 +} \left\{ \frac{2 ^{t} - 1}{t} - \frac{\left( \frac{1}{2} \right) ^{t} - 1}{t} \right.$ $\displaystyle = \ln 2 - \ln {\frac{1}{2}} = 2 \ln 2$ [참고] $a > 0$일 때, $a ^{t} - 1 = s$로 놓으면 $a ^{t} = s + 1$이므로 $\displaystyle t = \frac{\ln ( s + 1 )}{\ln a}$ $( a \ne 1 )$ $t \rightarrow 0$일 때, $s \rightarrow 0$이므로 $\displaystyle \lim\limits _{t \rightarrow 0} {} \frac{a ^{t} - 1}{t} = \lim\limits _{s \rightarrow 0} {\frac{s \ln a}{\ln ( s + 1 )}}$ $\displaystyle = \lim\limits _{s \rightarrow 0} {\frac{\ln a}{\frac{\ln ( s + 1 )}{s}}} = \ln a$