The limit $\lim_{n \to \infty}\left(2 + 1/n\right)^{n}$ How can one show that
$\displaystyle{\lim_{n \to \infty}\left(2 + {1 \over n}\right)^n}$ ?.

I am trying to shoehorn the definition of $e$ somewhere but fruitlessly so.
$$\displaystyle \lim_{n \to \infty} \left(2 + \frac 1n\right)^n\ge\lim_{n \to \infty} 2^n=+\infty$$
Applying L'Hopital $\lim(2+\frac{1}{n})^n=\lim(e^{\log(2+\frac{1}{n})n})=e^{\lim(\log(2+\frac{1}{n})/n^{-1})}$  the exponent will blow up.