As shown in the figure, in a right triangle ABC with $\overline { \mathrm { AB } } = 1 , \angle \mathrm { B } = \frac { \pi } { 2 }$, let D be the intersection of the angle bisector of $\angle \mathrm { C }$ and segment AB, and let E be the intersection of the circle with center A and radius $\overline { \mathrm { AD } }$ and segment AC. When $\angle \mathrm { A } = \theta$, let $S ( \theta )$ be the area of sector ADE and $T ( \theta )$ be the area of triangle BCE. What is the value of $\lim _ { \theta \rightarrow 0 + } \frac { \{ S ( \theta ) \} ^ { 2 } } { T ( \theta ) }$? [4 points] [Figure] (1) $\frac { 1 } { 4 }$ (2) $\frac { 1 } { 2 }$ (3) $\frac { 3 } { 4 }$ (4) 1 (5) $\frac { 5 } { 4 }$
As shown in the figure, in a right triangle ABC with $\overline { \mathrm { AB } } = 1 , \angle \mathrm { B } = \frac { \pi } { 2 }$, let D be the intersection of the angle bisector of $\angle \mathrm { C }$ and segment AB, and let E be the intersection of the circle with center A and radius $\overline { \mathrm { AD } }$ and segment AC. When $\angle \mathrm { A } = \theta$, let $S ( \theta )$ be the area of sector ADE and $T ( \theta )$ be the area of triangle BCE. What is the value of $\lim _ { \theta \rightarrow 0 + } \frac { \{ S ( \theta ) \} ^ { 2 } } { T ( \theta ) }$? [4 points]\\
\includegraphics[max width=\textwidth, alt={}, center]{ce47b871-6f7b-4821-9280-2ee0676a822e-07_350_402_754_1298}\\
(1) $\frac { 1 } { 4 }$\\
(2) $\frac { 1 } { 2 }$\\
(3) $\frac { 3 } { 4 }$\\
(4) 1\\
(5) $\frac { 5 } { 4 }$