lim⁡ln(1-cosx)/(lntgx) (х стремится к 0 )

правила форума
примеры
примеры

lim (x->0) ln (1 - cos x)/ln tg x = lim (x->0) (ln (1 - cos x))'/(ln tg x)' =
= lim (x->0) (sin x/(1 - cos x))/(1/cos^2 x * 1/tg x) =
= lim (x->0) (sin x/(1 - cos x))/(1/(sin x * cos x)) =
= lim (x->0) (sin^2 x * cos x)/(1 - cos x) = cos 0 * lim (x->0) sin^2 x/(1 - cos x) =
= lim (x->0) sin^2 x/(1 - (1 - 2 * sin^2 (x/2))) =
= lim (x->0) sin^2 x/(2 * sin^2 (x/2)) = lim (x->0) x^2/(2 * (x/2)^2) = lim (x->0) x^2/(x^2/2) = 2.