Question

一道极限题，高等数学，要详细过程，非诚勿扰

Answer 1

lim(x->0) {sin(e^x -1) - [e^(sinx) -1] }/x^4
=lim(x->0) [-(1/12) x^4]/x^4
=-1/12

x->0
e^x -1 ~ x+ (1/2)x^2 + (1/6)x^3 + (1/24)x^4

sin(e^x -1)
~ [x+ (1/2)x^2 + (1/6)x^3 + (1/24)x^4]
-(1/6)[x+ (1/2)x^2 + (1/6)x^3 + (1/24)x^4]^3
~ x+ (1/2)x^2 + (1/6)x^3 + (1/24)x^4 - (1/6)x^3 - (1/4)x^4
= x+ (1/2)x^2 - (5/24)x^4

e^(sinx) -1
~ [x - (1/6)x^3] + (1/2)[x- (1/6)x^3]^2 +(1/6)[x- (1/6)x^3]^3
+(1/24)[x- (1/6)x^3]^4
~ x - (1/6)x^3 + (1/2)x^2 - (1/6)x^4 +(1/6)x^3 +(1/24)x^4
= x +(1/2)x^2 -(1/8)x^4

sin(e^x -1) - [e^(sinx) -1]
~ - (5/24)x^4 +(1/8)x^4
=-(1/12)x^4

Answer 2

求导4次会出现一个常数，然后再把极限代入便可，计算量较大。

Answer 3

我知道x4次方用洛必达法则求导四次分母一定会变成一个数字，所以我知道这道题一定能做出来，接下来就是毅力了，算一算就能做出来，简便方法还不知道

Answer 4

这个题目，我刚才做了，计算量大

追答

不能用等价无穷小，要用洛必达法则

需要做吗？？？