lim sinx^x(x趋近于0+)求极限

直接求比较困难,考查其对数的极限.
设辅助函数g(x) = ln( (sin x)^x ) = x ln (sin x) = ln (sin x) / (1/x)
当x -> 0+时,这是∞/∞型不定式,连续使用罗比达法则,并利用sin x和x在x->0时为等价无穷小,得到
lim g(x) = lim ln(sin x) / (1/x) = lim cot(x) / (-1/x^2) = lim (-csc(x)^2) / (2/x^3) = -lim x^3 / (2( sin(x))^2)
= 0
从而原极限为e^0 = 1.