Question

函数求极限

函数求极限考研数学3的一道题见图片，求指导

Answer 1

为什么出错，原因是分母是x^4, 分子一定要展开到x^4才行
x->0
cosx ~ 1 - (1/2)x^2 + (1/24)x^4
//
x->0
sinx ~ x -(1/6)x^3
sin4x ~ 4x - (32/3)x^3
2x - (1/2)sin4x ~ (16/3)x^3
/
lim(x->0) [1/(sinx)^2 - (cosx)^2 /x^2]
=lim(x->0) [ x^2 - (sinx)^2.(cosx)^2]/[x^2.(sinx)^2]
=lim(x->0) [ x^2 - (1/4)(sin2x)^2]/x^4 (0/0)
分子，分母分别求导
=lim(x->0) [ 2x - sin2x.cos2x ]/(4x^3)
=lim(x->0) [ 2x - (1/2)sin4x ]/(4x^3)
=lim(x->0) [ (16/3)x^3 ]/(4x^3)
=4/3