What is the limit of the fraction tan4x/tan2x, x-->0 ?

We'll create the remarcable
limits:

lim tan x/x = 1, if
x->0

We'll re-write the
function:

lim [4x*(tan 4x)/4x]*[(2x)/2x*tan 2x] = lim
4x*lim [(tan 4x)/4x]*lim[(2x)/tan 2x]*lim (1/2x)

We know
that lim [(tan 4x)/4x] = 1 and lim[(2x)/tan 2x] = 1

lim
[4x*(tan 4x)/4x]*[(2x)/2x*tan 2x] = lim 4x*lim (1/2x)

lim
[4x*(tan 4x)/4x]*[(2x)/2x*tan 2x] =
(4/2)lim(x/x)

The limit of the given function
is : lim tan4x/tan2x = 2, if x -> 0.