Question

what is (cos(4x)-1)/x

Answer 1

We can use the special limit:

lim (x-->0) [1 - cos(x)]/x.
(You can easily derive this from lim (x-->0) sin(x)/x = 1 if needed)

Then, we have:

lim (x-->0) [1 - cos(4x)]/x
= 4 * lim (x-->0) [1 - cos(4x)]/(4x)
= 4 * 0, from the above limit
= 0.