lim cot x(1/sin x-1/x)

lim(x--->0) cot x(1/sin x-1/x)
=lim(x--->0)(cosx/sinx)*(x-sinx)/(xsinx)
=lim(x--->0)cosx(x-sinx)/(x(sinx)^2)
=lim(x--->0)(cosx(1-cosx)-sinx(x-sinx))/((sinx)^2+x*2sinxcosx)
=lim(x--->0)(cosx-(cosx)^2+(sinx)^2-xsinx)/((1-cos2x)/2+xsin2x)
=lim(x--->0)(cosx-xsinx-cos2x)/(1/2-1/2*cos2x+xsin2x)
=lim(x--->0)(-sinx-sinx-xcosx+2sin2x)/(sin2x+sin2x+2xcos2x)
=lim(x--->0)(-2sinx-xcosx+2sin2x)/(2sin2x+2xcos2x)
=lim(x--->0)(-2cosx-cosx+xsinx+4cos2x)/(4cos2x+2cos2x-4xsin2x)
=lim(x--->0)(-3cosx+xsinx+4cos2x)/(6cos2x-4xsin2x)
=(-3cos0+0+4cos0)/(6cos0-4*0)
=1/6