求极限 lim(x趋于0)[sinx/x]^(1/x^2)
问题描述：
求极限 lim(x趋于0)[sinx/x]^(1/x^2)
急,如题
答
lim[sinx/x]^(1/x²)
x→0
=lim[(x+sinx-x)/x]^(1/x²)
x→0
=lim[1+(sinx-x)/x]^{[(x/sinx-x)(sinx-x)/x](1/x²)}
x→0
=lim e^{[(sinx-x)/x](1/x²)}
x→0
=lim e^[(sinx-x)/x³]
x→0
=lim e^[(cosx-1)/3x²]
x→0
=lim e^[-sinx/6x]
x→0
=e^(-1/6)