一道利用等价无穷小求极限的题
来源:学生作业帮 编辑:百度作业网作业帮 分类:数学作业 时间:2024/07/31 06:18:05
一道利用等价无穷小求极限的题
lim[sinx+(x^2)sin(1/x)]/[(1+cosx)ln(1+x)]
x趋近于0
lim[sinx+(x^2)sin(1/x)]/[(1+cosx)ln(1+x)]
x趋近于0
![一道利用等价无穷小求极限的题](/uploads/image/z/16069126-22-6.jpg?t=%E4%B8%80%E9%81%93%E5%88%A9%E7%94%A8%E7%AD%89%E4%BB%B7%E6%97%A0%E7%A9%B7%E5%B0%8F%E6%B1%82%E6%9E%81%E9%99%90%E7%9A%84%E9%A2%98)
分子分母同除 x :
lim(x->0) [sinx+(x^2)sin(1/x)]/[(1+cosx)ln(1+x)]
=lim(x->0) [sinx /x + xsin(1/x)]/[(1+cosx){ln(1+x) /x}]
∵ lim(x->0) sinx /x = 1 ; lim(x->0) xsin(1/x) = 0
lim(x->0) (1+cosx) = 2 ; lim(x->0) ln(1+x) /x = 1
=[1+0]/2*1
= 1/2
lim(x->0) [sinx+(x^2)sin(1/x)]/[(1+cosx)ln(1+x)]
=lim(x->0) [sinx /x + xsin(1/x)]/[(1+cosx){ln(1+x) /x}]
∵ lim(x->0) sinx /x = 1 ; lim(x->0) xsin(1/x) = 0
lim(x->0) (1+cosx) = 2 ; lim(x->0) ln(1+x) /x = 1
=[1+0]/2*1
= 1/2