Определите предел функции:[latex] \lim_{x \to 0} (ln*ctg(x))^{tg(2x)} [/latex]

По формуле [latex]log _{a}x ^{k}=klog _{a}x,x>0 , [/latex] [latex] \lim_{x \to 0} tg2x\cdot(ln(ctgx))= \lim_{x \to 0} \frac{ln(ctgx)}{ctg2x}=( \frac{\infty}{\infty})= \\ = \lim_{x \to 0} \frac{(ln(ctgx))`}{(ctg2x)`}= \lim_{x \to 0} \frac{ \frac{1}{ctgx}(ctgx)` }{ \frac{1}{sin ^{2}2x }\cdot 2 }= \lim_{x \to 0} \frac{sin ^{2}2x }{2ctgx\cdot sin ^{2}x }= \\ = \lim_{x \to 0} \frac{4sin ^{2}x\cdot cos ^{2}x }{2sinx\cdot cosx} =0 [/latex]