lim(x趋向0)arctanx-sinx/x^3

来源：学生作业帮助网编辑：作业帮时间：2024/11/05 22:58:27

lim(x趋向0)arctanx-sinx/x^3
lim(x趋向0)arctanx-sinx/x^3

lim(x趋向0)arctanx-sinx/x^3
lim(x→0) (arctanx - sinx)/x³,洛必达法则
= lim(x→0) [1/(1 + x²) - cosx]/(3x²)
= (1/3)lim(x→0) (1 - cosx - x²cosx)/(x² + x⁴),洛必达法则
= (1/3)(1/2)lim(x→0) (sinx - 2xcosx + x²sinx)/(x + 2x³),洛必达法则
= (1/6)lim(x→0) (- cosx + 4xsinx + x²cosx)/(1 + 6x²)
= (1/6)(- 1 + 0)/(1 + 0)
= - 1/6

lim(x→0-）（arctanx-sinx/x^3)
=lim(x→0-）arctanx-lim(x→0-）sinx/x^3
=0-∞
=-∞
lim(x→0+）（arctanx-sinx/x^3)
=lim(x→0+）arctanx-lim(x→0+）sinx/x^3
=0-∞
=-∞
故此式极限不存在

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