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lim(x->0)(arctanx-sinx)/x^3=?
题目详情
lim(x->0) (arctanx-sinx)/x^3=?
▼优质解答
答案和解析
利用洛必达法则
lim(x→0) (arctanx - sinx)/x³
= lim(x→0) [1/(1 + x²) - cosx]/(3x²)
= (1/3)lim(x→0) (1 - cosx - x²cosx)/(x² + x⁴)
= (1/3)(1/2)lim(x→0) (sinx - 2xcosx + x²sinx)/(x + 2x³)
= (1/6)lim(x→0) (- cosx + 4xsinx + x²cosx)/(1 + 6x²)
= (1/6)(- 1 + 0)/(1 + 0)
= - 1/6
lim(x→0) (arctanx - sinx)/x³
= lim(x→0) [1/(1 + x²) - cosx]/(3x²)
= (1/3)lim(x→0) (1 - cosx - x²cosx)/(x² + x⁴)
= (1/3)(1/2)lim(x→0) (sinx - 2xcosx + x²sinx)/(x + 2x³)
= (1/6)lim(x→0) (- cosx + 4xsinx + x²cosx)/(1 + 6x²)
= (1/6)(- 1 + 0)/(1 + 0)
= - 1/6
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