JEE Math Practice Question

Let the solution curve $$y=y(x)$$ of the differential equation $$\left(1+\mathrm{e}^{2 x}\right)\left(\frac{\mathrm{d} y}{\mathrm{~d} x}+y\right)=1$$ pass through the point $$\left(0, \frac{\pi}{2}\right)$$. Then, $$\lim\limits_{x \rightarrow \infty} \mathrm{e}^{x} y(x)$$ is equal to :

1. A.$$ \frac{\pi}{4} $$
2. B.$$ \frac{3\pi}{4} $$
3. C.$$ \frac{\pi}{2} $$
4. D.$$ \frac{3\pi}{2} $$