Question

函数求极限问题

如图为什么不能这么做

Answer 1

y->0+

(1+2y+y^3)^(1/3) = 1+(2/3)y +o(y)

e^y = 1+y+o(y)

(1+2y+y^3)^(1/3) -e^y = -(1/3)y +o(y)

//

lim(x->+无穷) [(x^3+2x^2+1)^(1/3) -xe^(1/x) ]

y=1/x

=lim(y->0+) [(1/y^3+2/y^2+1)^(1/3)  -e^y/y ]

通分母

=lim(y->0+) [(1+2y+y^3)^(1/3)  -e^y]/y 

=lim(y->0+) -(1/3)y/y 

=-1/3

Answer 2

最后一步局部（分母）求极限 错误。
lim<t→0+>[(1+2t+t^3)^(1/3) - e^t]/t
= lim<t→0+>[(1+2t+t^3)^(1/3)-1]/t - lim<t→0+>( e^t-1)/t
= 2/3 - 1 = -1/3