Limits and trigonometry

Okay. So I have this homework problem here and I can't really figure out how to do it. The problem is...

The limit of sin(2x)/x as x approaches 0.

I know how to do it using my TI-83, but I'm supposed to be able to do it by hand using these four facts:

The limit of sin(x)/x as x approaches zero = 1
The limit of (cos(x)-1)/x as x approaches zero = 0
The limit of sin(x) as x approaches zero = 0
The limit of cos(x) as x approaches zero = 1

I've tried rewriting it using the double angle formula thing to get...

(2sin(x)cos(x))/x

Which I tried rewriting as (2/x)+(sin(x)/x)+(cos(x)/x)

But then I get all confused and I don't know what to do. Help?

if x on the bottom have to divide by zero and u cant so trick question cant answer this or id say 0