作业帮高数：一道极限题 cos(sinx)
来源：学生作业帮 编辑：拍题作业网作业帮 分类：数学作业 时间：2024/04/24 11:37:10
高数：一道极限题 cos(sinx)
lim[cos(sinx)-cosx]/x^4
cosx=1-(1/2)x^2+(1/24)x^4+o(x^4)
sinx=x-(1/6)x^3+o(x^3)
cos(sinx)=1-(1/2)(x-(1/6)x^3)^2+(1/24)x^4+o(x^4)
=1-(1/2)x^2+(1/6)x^4+(1/24)x^4+o(x^4)
原式=lim[1-(1/2)x^2+(1/6)x^4+(1/24)x^4-(1-(1/2)x^2+(1/24)x^4)+o(x^4)]/x^4
=1/6
cosx=1-(1/2)x^2+(1/24)x^4+o(x^4)
sinx=x-(1/6)x^3+o(x^3)
cos(sinx)=1-(1/2)(x-(1/6)x^3)^2+(1/24)x^4+o(x^4)
=1-(1/2)x^2+(1/6)x^4+(1/24)x^4+o(x^4)
原式=lim[1-(1/2)x^2+(1/6)x^4+(1/24)x^4-(1-(1/2)x^2+(1/24)x^4)+o(x^4)]/x^4
=1/6