접하는 원의 반지름 극한

양의 실수 $t$에 대하여 점 $\mathrm{P} \mathit{_{n}} \left( t , f \left( t \right) \right)$라 하면 $\displaystyle f' \left( t \right) = \frac{f \left( t \right)}{t}$, $\displaystyle \frac{12 t ^{3}}{n ^{3}} = \frac{4 t ^{3}}{n ^{3}} + 1$, $\displaystyle t ^{3} = \frac{n ^{3}}{8}$, $\displaystyle t = \frac{n}{2}$ $\displaystyle \mathrm{P} \mathit{_{n}} \left( \frac{n}{2} , \frac{3}{2} \right)$이므로 직선 $l _{n}$의 방정식은 $\displaystyle y = \frac{3}{n} x$ 원 $C _{n}$의 중심을 ${\mathrm{C}}$라 하고 두 점 $\mathrm{P} \mathit{_{n}}$, ${\mathrm{C}}$에서 $x$축에 내린 수선의 발을 각각 $\mathrm{Q} \mathit{_{n}}$, $\mathrm{R} \mathit{_{n}}$이라 하자. 점 ${\mathrm{C}}$에서 선분 $\mathrm{P} \mathit{_{n}} \mathrm{Q} \mathit{_{n}}$에 내린 수선의 발을 ${\mathrm{H}} \mathit{_{n}}$이라 하자. $\displaystyle \angle \mathrm{CP} \mathit{_{n}} \mathrm{O} \mathit{=} \angle \mathrm{OQ} \mathit{_{n}} \mathrm{P} \mathit{_{n}} = \frac{\pi}{2}$이므로 $\angle \mathrm{P} \mathit{_{n}} \mathrm{OQ} \mathit{_{n}} = \angle \mathrm{CP} \mathit{_{n}} \mathrm{H} \mathit{_{n}}$

$\displaystyle {\overline{\mathrm{OP} \mathit{_{n}}}} = \sqrt{\left( \frac{n}{2} \right) ^{2} + \left( \frac{3}{2} \right) ^{2}} = \frac{\sqrt{n ^{2} + 9}}{2}$이고, $\angle \mathrm{P} \mathit{_{n}} \mathrm{OQ} \mathit{_{n}} = \theta$라 하면 $\displaystyle \cos \theta = \frac{{\overline{\mathrm{OQ} \mathit{_{n}}}}}{{\overline{\mathrm{OP} \mathit{_{n}}}}} = \frac{n}{\sqrt{n ^{2} + 9}}$ $\displaystyle {\overline{\mathrm{P} \mathit{_{n}} \mathrm{C}}} = {\overline{\mathrm{CR} \mathit{_{n}}}} = {\overline{\mathrm{H} \mathit{_{n}} \mathrm{Q} \mathit{_{n}}}} = r _{n}$, $\displaystyle {\overline{\mathrm{P} \mathit{_{n}} \mathrm{Q} \mathit{_{n}}}} = \frac{3}{2}$이므로 $\displaystyle {\overline{\mathrm{P} \mathit{_{n}} \mathrm{Q} \mathit{_{n}}}} = {\overline{\mathrm{P} \mathit{_{n}} \mathrm{H} \mathit{_{n}}}} + {\overline{\mathrm{H} \mathit{_{n}} \mathrm{Q} \mathit{_{n}}}} = r _{n} \times \cos \theta + r _{n} = \frac{3}{2}$ $r _{n}$$\displaystyle = \frac{3}{2 \left( 1 + \cos \theta \right)}$$\displaystyle = \frac{3 \sqrt{n ^{2} + 9}}{2 \left( \sqrt{n ^{2} + 9} + n \right)}$ $\displaystyle \lim\limits _{n \rightarrow \infty} {} n ^{2} \left( 4 r _{n} - 3 \right)$$\displaystyle = \lim\limits _{n \rightarrow \infty} {} n ^{2} \times \left( \frac{6 \sqrt{n ^{2} + 9}}{\sqrt{n ^{2} + 9} + n} - 3 \right)$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} n ^{2} \left( \frac{3 \sqrt{n ^{2} + 9} - 3 n}{\sqrt{n ^{2} + 9} + n} \right)$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} 3 n ^{2} \left\{ \frac{9}{\left( \sqrt{n ^{2} + 9} + n \right) ^{2}} \right\}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{27}{\left( \sqrt{1 + \frac{9}{n ^{2}}} + 1 \right) ^{2}} = \frac{27}{4}$ 이므로 $\displaystyle 40 \times \lim\limits _{n \rightarrow \infty} {} n ^{2} \left( 4 r _{n} - 3 \right) = 270$