=\ldots=e^{\diplaystyle \lim_{x \to +\infty}{\frac{\ln\left(1+\frac{x+1}{x^2+2x+3}\right)}{\frac{x+1}{x^2+2x+3}}\cdot \frac{(x+1)x}{x^2+2x+3} \cdot \frac{1}{2+\frac{\ln\left(1+\frac{2x+3}{x^2}\right)}{\ln{x}}}}}=\sqrt{e}