포물선 위 각의 극한

점 $\mathrm{P}$에서 $x$축에 내린 수선의 발을 $\mathrm{{H}}$, $\angle PAH = \alpha , \angle POH = \beta , \angle APO = \theta$라 하면 $\alpha + \beta + \theta = \pi$ 점 $\mathrm{P}$를 $\left( t , t ( t + 1 ) \right)$라 두면 점 $\mathrm{H}$는 $\left( t , 0 \right)$ ㄱ) $\displaystyle \tan \alpha = \frac{PH}{AH} = \left| \frac{t ( t + 1 )}{t + 1} \right| = \left| t \right|$ $\displaystyle \lim\limits _{t \rightarrow 0} {} \tan \alpha = 0$ , $\displaystyle \therefore \lim\limits _{t \rightarrow 0} {} \alpha = 0$ ㄴ) $\displaystyle \tan \beta = \frac{PH}{OH} = \left| \frac{t ( t + 1 )}{- t} \right| = \left| t + 1 \right|$ $\displaystyle \lim\limits _{t \rightarrow 0} {} \tan \alpha = 1$ , $\displaystyle \therefore \lim\limits _{t \rightarrow 0} {} \beta = \frac{\pi}{4}$ ㄱ)ㄴ)에서 $\displaystyle \lim\limits _{t \rightarrow 0} {} \theta = \lim\limits _{t \rightarrow 0} {} \left( \pi - \alpha - \beta \right) = \pi - 0 - \frac{\pi}{4} = \frac{3 \pi}{4} = 135 ^\circ$ [별해] 점 $P$의 좌표를 $P ( a , a ^{2} + a ) , \angle APO = \theta$라 하면 $\displaystyle \overline{AP} = \sqrt{( a + 1 ) ^{2} + ( a ^{2} + a ) ^{2}} = ( a + 1 ) \sqrt{1 + a ^{2}}$ $\displaystyle \overline{OP} = \sqrt{a ^{2} + ( a ^{2} + a ) ^{2}} = - a \sqrt{2 + 2 a + a ^{2}}$ (단, $- 1 < a < 0$) $\displaystyle \therefore \cos \theta = \frac{( a + 1 ) ^{2} ( 1 + a ^{2} ) + a ^{2} ( 2 + 2 a + a ^{2} ) - 1}{2 ( a + 1 ) \sqrt{1 + a ^{2}} ( - a ) \sqrt{2 + 2 a + a ^{2}}}$ $\displaystyle = \frac{2 a ( a ^{3} + 2 a ^{2} + 3 a + 1 )}{- 2 a ( a + 1 ) \sqrt{1 + a ^{2}} \sqrt{2 + 2 a + a ^{2}}}$ 이고 $P \rightarrow 0$일 때 $a \rightarrow 0$ $\displaystyle \lim\limits _{a \rightarrow 0} {} \cos \theta = - \frac{1}{\sqrt{2}}$에서 $\theta = 135 ^\circ$