用等价无穷小量代换求极限

先化简，利用分子，

分母有理化，

然后用等价无穷小量代换，

就可以求出极限为4/3.

具体解答如图所示

原式=lim(x->0) [(1/3)*xe^x]*x/[1-√(1+cosx-1)]

=lim(x->0) [(1/3)*x^2]/[-(1/2)*(cosx-1)]
=(2/3)*lim(x->0) (x^2)/[(x^2)/2]
=4/3

x->0
分子
xe^x =x +o(x)

(1+xe^x)^(1/3) = [ 1+ x+o(x) ] ^(1/3) = 1+ (1/3)x +o(x)
(1+xe^x)^(1/3) -1 =(1/3)x +o(x)
x.[(1+xe^x)^(1/3) -1 ]=(1/3)x^2 +o(x^2)
分母

cosx = 1-(1/2)x^2 +o(x^2)
√cosx = √[1-(1/2)x^2 +o(x^2)] = 1 -(1/4)x^2 +o(x^2)
1-√cosx =(1/4)x^2 +o(x^2)
lim(x->0) x.[ (1+xe^x)^(1/3) -1 ] /[1- √cosx]
=lim(x->0) (1/3)x^2 /[(1/4)x^2]
=4/3