지수함수 극한 도형

[출제의도] 지수함수의 극한 이해하기 두 점 ${\mathrm{P}}$, ${\mathrm{Q}}$의 좌표는 각각 $\mathrm{P} \mathit{\left( t , e ^{2 t} - 1 \right)}$, $\mathrm{Q} \mathit{\left( f \left( t \right) , 0 \right)}$ $\displaystyle {\overline{\mathrm{PQ}}} ^{2} = {\overline{\mathrm{OQ}}} ^{2}$이므로 $\left\{ f \left( t \right) - t \right\} ^{2} + \left( e ^{2 t} - 1 \right) ^{2} = \left\{ f \left( t \right) \right\} ^{2}$에서 $f \left( t \right)$$\displaystyle = \frac{t}{2} + \frac{\left( e ^{2 t} - 1 \right) ^{2}}{2 t}$ $\displaystyle \lim\limits _{t \rightarrow 0 +} {} \frac{f \left( t \right)}{t}$$\displaystyle = \lim\limits _{t \rightarrow 0 +} {} \left\{ \frac{1}{2} + \frac{1}{2} \left( \frac{e ^{2 t} - 1}{t} \right) ^{2} \right\}$ $\displaystyle = \lim\limits _{t \rightarrow 0 +} {} \left\{ \frac{1}{2} + \frac{1}{2} \left( \frac{e ^{2 t} - 1}{2 t} \times 2 \right) ^{2} \right\}$ $\displaystyle = \frac{1}{2} + \frac{1}{2} \times \left( 1 \times 2 \right) ^{2} = \frac{5}{2}$