先等价无穷小代换:
lim(x→ 0)(tanx-sinx)/xsinx^2
=lim(x→ 0)(tanx-sinx)/ x^3
原式=lim (sin/cosx - sinx)/x³
= lim sinx(1-cosx)/(x³cosx)
注意 x与sinx是等价无穷小
1-cosx 与 x²/2是等价无穷小【1-cosx=2sin²(x/2)~2*(x/2)²=x²/2】
所以
原式= lim (x * x²/2)/(x³cosx)
=lim 1/(2cosx)
=1/2
也可以:
lim[x→0] (tanx-sinx)/x³
=lim[x→0] (sinx/cosx-sinx)/x³
=lim[x→0] (sinx-sinxcosx)/(x³cosx)
=lim[x→0] sinx(1-cosx)/(x³cosx)
=lim[x→0] sin³x(1-cosx)/(x³sin²xcosx)
=lim[x→0] (sinx/x)³·(1-cosx)/(sin²xcosx)
=lim[x→0] (sinx/x)³·(1-cosx)/[(1-cos²x)cosx]
=lim[x→0] (sinx/x)³·(1-cosx)/[(1+cosx)(1-cosx)cosx]
=lim[x→0] (sinx/x)³·1/[(1+cosx)cosx]
=1·1/(1+1)
=1/2
lim(x→ 0)(tanx-sinx)/xsinx^2
lim(x→ 0)(tanx-sinx)/xsinx^2
数学人气:916 ℃时间:2020-01-28 20:12:30
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