Lim(cos x)^(ctg 2x/sin 3x) x- 2pi - вопрос №6818855232 от dan4ikqwerty 11.05.2020 09:19

Question

Алгебра: Lim(cos x)^(ctg 2x/sin 3x) x- 2pi

dan4ikqwerty · Answer

Lim(cos x)^(ctg 2x/sin 3x)=..., x-2pi=t t>0 x=t+2pi
x->2pi    x->2pi

=Lim(cos( t+2pi))^(ctg(2(t+2pi)/sin3(t+2pi)) =Lim(cos( t))^[ctg (2t)/sin 3(t)]=
t-->0 t-->0

=e^{Lim[ctg (2t)/sin 3(t)]·ln(cos t)}=e^{Lim[1/(2t·3t)]·ln[(cos t-1)+1]}=
t-->0 t-->0

=e^{Lim[1/(6t²)]·[cos t-1]}=e^{Lim[1/(6t²)]·[-2sin²(t/2)]}=e^{Lim[1/(6t²)]·[-t²/2)]}=
   t-->0    t-->0    t-->0
=e^{Lim[1/(6)]·[-1/2)]}=e^(-1/12)
   t-->0

Lim(cos x)^(ctg 2x/sin 3x) x-> 2pi