`lim_(x->0)cos^(2)((1-cos^2)(1-cos^2( .....cos^2(x))))/(sin[pi((sqrt(x+4)-2)/x)])`

Evaluate the following limits (if exists), where {,} denotes the fractional part of x and [,] denotes the greatest integer part lim_(x rarr 0) (cos^2(1-cos^2(1-cos^2.......cos^2(x)......)))/(sin[pi((sqrt(x+4) - 2)/x) ]

lim_(x->0)(1-cos x-cos2x+cos x*cos2x)/(x^(4))

lim_(x rarr0)((cos x)^((1)/(2))-(cos x)^((1)/(3)))/(sin^(2)x) is

lim_(x to 0) (cos 2x-1)/(cos x-1)

lim_(x to 0) (cos 2x-1)/(cos x-1)

lim_(xto(pi)) (1-sin(x/2))/(cos(x/2)(cos(x/4)-sin(x/4)))

lim_(x rarr0)(sin^(2)4x+2cos^(2)x-2cos x)/(cos^(2)x-cos^(3)x)

Lim_(x->0)(1-cos x-cos2x+cos x cos2x)/(x^(4))

lim_(x rarr0)(cos^(2)x-sin^(2)x-1)/(sqrt(x^(2)+4)-2)=

If lim_(x rarr 0)((1-cos((x^2)/2)-cos((x^2)/4)+cos((x^2)/2)cos((x^2)/4))/(x^8))=2^(-k) . Find k.