What is the limit of the function y given by y=(cos x-cos7x)/x, if x approaches to 0 value?

We'll substitute x by the value of accumulation point, in the
given expresison of function:

y = (cos 0 - cos 7*0)/0 = (1 - 1)/0 =
0/0

Since we've get an indetermination, we'll apply l'Hospital
rule:

lim (cos x - cos 7x)/x = lim (cos x - cos
7x)'/(x)'

lim (cos x - cos 7x)'/(x)' = lim (-sin x + 7 sin
7x)/1

We'll substitute x by accumulation
point:

lim (-sin x + 7 sin 7x) = -sin 0 + 7 sin
7*0

lim (-sin x + 7 sin 7x) = 0 -
7*0

lim (-sin x + 7 sin 7x) = 0

Therefore, for x->0, the limit
of the function is: lim (cos x - cos 7x)/x = 0.