Calculate limit of $(1 + \frac{1}{n^2})^n$

Question

How to calculate

$$ \lim_{n \to \infty}\left(1 + \frac{1}{n^2}\right)^n $$

using only the very basic limit features (I cannot use the fact that it equals to $e^0 = 1$)?

Answer 1

Recall Bernoulli's inequality $$e^x\ge x+1\quad \text{for all }x\in\Bbb R$$

Therefore $$\left(1+n^{-2}\right)^n\le \left(e^{n^{-2}}\right)^n=e^{1/n}$$