Let the solution curve \( y=y(x) \) of the differential equation \( (1+e^{2x})\!\left(\dfrac{dy}{dx}+y\right)=1 \) pass through the point \( \left(0,\dfrac{\pi}{2}\right) \). Then, \( \displaystyle\lim_{x\to \infty} e^{x}y(x) \) is equal to:
\( \dfrac{\pi}{4} \)
\( \dfrac{3\pi}{4} \)
\( \dfrac{\pi}{2} \)
\( \dfrac{3\pi}{2} \)
