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

I dont know how to do it without using continuity of exponential function
I mean I cannot do this:
$$\lim_{x\to\infty}a_n =a  \wedge  \lim_{x\to\infty}b_n =b\Rightarrow  \lim_{x\to\infty}{a_n}^{b_n} =a^b$$ 

Use Bernoulli:
$$\left(1+\frac{2}{n^2}\right)^n= \frac{1}{\left(\frac{n^2}{n^2+2}\right)^n}= \frac{1}{\left(1-\frac{2}{n^2+2}\right)^n}$$
And by Bernoulli
$$\left(1-\frac{2}{n^2+2}\right)^n \geq 1-\frac{2n}{n^2+2}=\frac{n^2-2n+2}{n^2+2}$$
Therefore
$$1 \leq \left(1+\frac{2}{n^2}\right)^n \leq \frac{n^2+2}{n^2-2n+2}$$
