lim(x→0)cotx[1/sinx-1/x]

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lim(x→0)cotx[1/sinx-1/x]

lim(x→0)cotx[1/sinx-1/x]
lim(x→0)cotx[1/sinx-1/x]

lim(x→0)cotx[1/sinx-1/x]
lim(x→0)cotx[1/sinx-1/x]
=lim(x→0)cotx(x-sinx)/(xsinx)
=lim(x→0)(x-sinx)/(xsinxtanx)
=lim(x→0)(x-sinx)/(x^3) (0/0)
=lim(x→0)(1-cosx)/(3x^2)
=lim(x→0)(x^2/2)/(3x^2)
=1/6

=lim(x->0)(cosx/sinx^2-cosx/xsinx)
=lim(x->0)(xcosx-sinxcosx)/x^2sinx^2)
=lim(x->0)(xcosx-(1/2)sin2x)/(x^2sinx^2)
=lim(x->0)(cosx-xsinx-cos2x)/(2xsinx^2+x^2sin2x)
=lim(x->0)(-2sinx-xcosx-2sin2x)/(2sinx^2+4xsin2x+2x^2cos2x)
=lim(x->0)(-3cosx+xsinx-4cos2x)/(6sin2x+12xcos2x+4x^2sin2x)
=-∝