Limit of (1-2/n)^n as n --> Infinity

[wimma]

Homework Statement

Find the limit as n--> infinity of (1-2/n)^n

Homework Equations

We know (1+1/x)^x --> e as n--> infinity

The Attempt at a Solution

I worked it out as e^(-2) using log but I can't get it out using the fundamental limit above. I know it's the square of (1-1/x)^x (where we let x=n/2), just I don't know how to show that (1-1/x)^x --> 1/e. If you could let x |--> -x somehow I'd get the desired result using the limit laws but I'm not sure that's allowed.

 

[Mark44]

Let n = -2x. This makes your limit
$$\lim_{-2x \to \infty} (1 + \frac{1}{x})^{-2x}$$

With a bit of adjustment you can use the limit you know.

 

[wimma]

but won't the parameter go to -infinity so we can't equate (1+1/x)^x to e?

 

[Mark44]

As it turns out,
$$\lim_{x \to -\infty} (1 + \frac{1}{x})^x~=~e$$

Can you use this fact?