Question

麻烦帮求下这个极限的值X趋于0 求X∧3分之（tanX-sinX）的极限的值

Answer 1

(tanx-sinx)/x³

=(sinx/cosx-sinx)/x³

=(sinx/x)*(1-cosx)/x²cosx

=(sinx/x)*[1-(1-2sin²x/2)]/x²cosx

=(sinx/x)*2sin²(x/2)/x²cosx

=(sinx/x)*2sin²(x/2)/4(x/2)²cosx

=(1/2)*(sinx/x)*[sin(x/2)/(x/2)]²*(1/cosx)

后面每项极限是1

所以极限=1/2

或者tanx-sinx=sinx(1/cosx-1)

(tanx-sinx)/(sinx)^3=(1/cosx-1)/(sinx)^2

用洛必达法则,分子分母分别求导

=(1/cosx-1)'/[(sinx)^2]'

=(sinx/(cosx)^2)/(2sinxcosx)

=1/[2(cosx)^3]

=1/2

(cos0=1)