How to evaluate the limit of (cos x - cos 3x) / x*sin x if x-->0 ?

Since the trigonometric functions from numerator are
matching, we'll transform the difference into a
product:

cos x - cos 3x = 2[sin
(x+3x)/2]*[sin(3x-x)/2]

cos x - cos 3x = 2 sin2x *sin
x

We'll re-write the
fraction:

(cos x - cos 3x) / x*sin x = 2 sin2x *sin x/x*sin
x

We'll simplify and we'll
get:

2 sin2x *sin x/x*sin x = 2
sin2x/x

Now, we'll evaluate the
limit:

lim (cos x - cos 3x) / x*sin x = lim 2
sin2x/x

We'll create the remarcable
limit:

lim 2 sin2x/x = 2 lim
(sin2x/2x)*2

2 lim (sin2x/2x)*2 = 4 lim
(sin2x/2x)

But lim (sin2x/2x)  =1, if x ->
0

lim (cos x - cos 3x) / x*sin x =
4