1. ## infinity question

How do I show that as $\displaystyle n\to\infty$ of $\displaystyle (1-\frac{x}{\theta n})^n$ = $\displaystyle e^{-\frac{x}{\theta}}$??

2. Is it $\displaystyle \lim_{n \rightarrow \infty} (1+\frac{x}{n})^n$?

You can prove that the sequence has an upper bound (3) and a lower bound (2) using the binomial expansion. I guess that's where to start!

3. Hello,

Note that :

$\displaystyle \left(1-\frac{x}{n \theta}\right)^n=\left(\left(1-\frac{x}{n \theta}\right)^{-n\theta/x}\right)^{-x/\theta}$

Substitute $\displaystyle \frac 1t=-\frac{x}{n \theta}$ and you will get :

$\displaystyle \lim_{t \to \infty} \left(\left(1+\frac 1t\right)^t\right)^{-x/\theta}$

and this is a known formula