求 lim(tan x-sin x)/(sin x)^3 x趋于0的极限值详细步骤
问题描述：
求 lim(tan x-sin x)/(sin x)^3 x趋于0的极限值
详细步骤
答
0.5
答
原式= lim(tan x-sin x)/x^3
=lim sin x (1-cos x) /[cos x * x^3]
=lim 0.5 x^2 / x^2 (分析:(1-cos x)~ 0.5x^2; sinx ~ x; cosx=1)
=0.5
建议：在求极限时尽量使用等价无穷小化简，然后再考虑罗比达法则运算
答
原式=lim(x->0)[(sinx/cosx-sinx)/sin³x]
=lim(x->0)[(1-cosx)/(sin²xcosx)]
=lim(x->0)[2sin²(x/2)/(sin²xcosx)]
=lim(x->0)[(x/sinx)²*(sin(x/2)/(x/2))²*(1/(2cosx))]
=1*1*(1/2) (利用重要极限lim(x->0)(sinx/x)=1)
=1/2