作业帮lim趋于0时 (xsinx-sinx^2)/x^4
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/03 01:02:28
lim趋于0时 (xsinx-sinx^2)/x^4
lim趋于0时 (xsinx-sinx^2)/x^4
lim趋于0时 (xsinx-sinx^2)/x^4
lim(x->0)(xsinx-sinx^2)/x^4
=lim(x->0) (sinx+xcosx-2xcosx^2)/4x^3
=lim(x->0)(cosx+cosx-xsinx-2cosx^2+2x^2sinx^2)/12x^2
=lim(x->0)(-2sinx-sinx-xcosx+2xsinx^2+4xsinx^2+4x^3cosx^2)/24x
=lim(x->0)(-3cosx-cosx+xsinx+2sinx^2+2xcosx^2+4sinx^2+8x^2cosx^2+12x^2cosx^2-8x^4sinx^2)/24
=-4/24=-1/6
1
罗比达法则，-1/24
原式=lim(x->0){[x(x-x³/3+x^5/15+o(x^6))-(x²-x^6/3+o(x^6))]/x^4}
(展开成级数，o(x)表示高阶无穷小)
=lim(x->0){[(x²-x^4/3+x^6/15+o(x^6))-(x²-x^6/3+o(x^6))...
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原式=lim(x->0){[x(x-x³/3+x^5/15+o(x^6))-(x²-x^6/3+o(x^6))]/x^4}
(展开成级数，o(x)表示高阶无穷小)
=lim(x->0){[(x²-x^4/3+x^6/15+o(x^6))-(x²-x^6/3+o(x^6))]/x^4}
=lim(x->0){[-x^4/3+(1/15-1/3)x^6+o(x^6)]/x^4}
=lim(x->0)[-1/3+(1/15-1/3)x²+o(x²)]
=-1/3
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