Question

当X趋近于0时，lim（tanx-sinx）除以limx的三次方等于多少

Answer 1

就是用最直接最简单的方法做,下面提供2种不同的方法:参考定理lim[x→0] sinx/x=1 lim[x→0] (tanx-sinx)/x�0�6
=lim[x→0] (sinx/cosx-sinx)/x�0�6
=lim[x→0] (sinx-sinxcosx)/(x�0�6cosx)
=lim[x→0] sinx(1-cosx)/(x�0�6cosx)
=lim[x→0] sin�0�6x(1-cosx)/(x�0�6sin�0�5xcosx)
=lim[x→0] (sinx/x)�0�6·(1-cosx)/(sin�0�5xcosx)
=lim[x→0] (sinx/x)�0�6·(1-cosx)/[(1-cos�0�5x)cosx]
=lim[x→0] (sinx/x)�0�6·(1-cosx)/[(1+cosx)(1-cosx)cosx]
=lim[x→0] (sinx/x)�0�6·1/[(1+cosx)cosx]
=1·1/(1+1)
=1/2 lim[x→0] (tanx-sinx)/x�0�6
=lim[x→0] (sinx/cosx-sinx)/x�0�6
=lim[x→0] (sinx/x)·(1-cosx)/(x�0�5cosx)
=lim[x→0] (sinx/x)·[1-(1-2sin�0�5x/2)]/(x�0�5cosx)
=lim[x→0] (sinx/x)·2sin�0�5(x/2)/(x�0�5cosx)
=lim[x→0] (sinx/x)·2sin�0�5(x/2)/[4(x/2)�0�5cosx]
=lim[x→0] (1/2)·(sinx/x)·[sin(x/2)/(x/2)]�0�5·(1/cosx)=1/2·1·1�0�5·1=1/2

Answer 2

用等价无穷小来解题