Calculate: $\lim\limits_{x \to \infty}\left(\frac{x^2+2x+3}{x^2+x+1} \right)^x$ How do I calculate the following limit without using l'Hôpital's rule?
$$\lim_{x \to \infty}\left(\frac{x^2+2x+3}{x^2+x+1} \right)^x$$

$$
\begin{aligned}
\lim_{x \to \infty}\left(\frac{x^2+2x+3}{x^2+x+1} \right)^x
& = \lim _{x\:\to \infty }\left(e^{x\ln\left(\frac{x^2+2x+3}{x^2+x+1}\right)}\right)
\\& \approx \lim _{x\:\to \infty }\left(e^{x\left(\frac{x^2+2x+3}{x^2+x+1}-1\right)}\right)
\\& = \lim _{x\:\to \infty }\left(e^{\frac{x^2+2x}{x^2+x+1}}\right)
\\& = \color{red}{e}
\end{aligned}
$$
