Question

求当x趋于0时 （sinx/x）^(1/(1-cosx))的极限

Answer 1

这是 (1 + 无穷小) ^ ∞ 类型
ln【（sinx/x）^(1/(1-cosx)) 】= 1/(1- cosx) * [ ln(sinx) - lnx ]
= [ ln(sinx) - lnx ] / (1- cosx)
lim [ [ ln(sinx) - lnx ] / (1- cosx),x->0 ]
= lim[ (cosx /sinx - 1/x ) / sinx ,x->0] = lim[ (x - tanx) / ( x * tanx * sinx,x->0 ]
= lim[ (x - tanx) / x³,x->0]
= lim[ ( 1- sec²x) / (3x²) ,x->0] = (-1/3) lim[ tan²x / x² ,x->0 ]
= (-1/3)
原式= e^(-1/3)