极限的计算1.limx-0(x-sinx)/x32.limx-01/xsinx/3x3是x的3次方

Question

极限的计算
1.limx-0 (x-sinx)/x3
2.limx-0 1/xsinx/3
x3是x的3次方

Answer 1

1、lim[x→0] (x-sinx)/x³
洛必达法则
=lim[x→0] (1-cosx)/(3x²)
等价无穷小代换,1-cosx与(1/2)x²等价
=lim[x→0] (1/2)x²/(3x²)
=1/6
2、lim[x→0] (1/x)sin(x/3)
等价无穷小代换,sin(x/3)与x/3等价
=lim[x→0] (1/x)(x/3)
=1/3