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 :