If `f (x) =lim _(x to oo) x ((3)/(2) + [cos x)(sqrt(n^(2) +1)-sqrt(n ^(2) -3n +1)))` where [y] denotes largest integer `le y,` then identify the correct statement (s).

If f(x)=(lim)_(nvecoo)(3/2+[cosx](sqrt(n^2+1)-sqrt(n^2-3n+1))) where [y] denotes largest integer lt=, then identify the correct statement(s). ("lim")_(nvecoo)f(x)=0 ("lim")_(nvecpi/2)f(x)=(3pi)/4 f(x)=(3x)/2AAx in [0,pi/2] f(x)=0AAx in (pi/2,(3pi)/2)

The value of lim_(nto oo)(sqrt(n^(2)+n+1)-[sqrt(n^(2)+n+1)]) where [.] denotes the greatest integer function is

lim_(x to oo ) (sqrt(3x^(2)-1)-sqrt(2x^(2)-3))/(4x+3)

Let intsec^(- 1)[-sin^2x]dx=f(x)+C where [y] denotes largest integer le y, then find the value of [f(8/(pix))]^('') at x=2.

lim _(x to oo) (1)/(n ^(3))(sqrt(n ^(2)+1)+2 sqrt(n ^(2) +2 ^(2))+ .... + n sqrt((n ^(2) + n ^(2)))=:

Let f(x)=lim_(n to oo) ((2 sin x)^(2n))/(3^(n)-(2 cos x)^(2n)), n in Z . Then

lim_(x->oo)x^(3/2)(sqrt(x^3+1)-sqrt(x^3-1))

lim_(n->oo) ((sqrt(n^2+n)-1)/n)^(2sqrt(n^2+n)-1)

lim_(n->oo)cos^2(pi(3sqrt(n^3+n^2+2n)-n)) where n is an integer,equals

f(x) = lim_(n->oo) sin^(2n)(pix)+[x+1/2] , where [.] denotes the greatest integer function, is