求当x趋向于0时,函数（1-三次根号(1-x+x²））/x的极限

题目

求当x趋向于0时,函数（1-三次根号(1-x+x²））/x的极限
要快,答案是1/3

答案

分子分母同乘：[ 1+(1-x+x²)^(1/3)+(1-x+x²)^(2/3) ] 有理化：
lim(x->0) [1-(1-x+x²)^(1/3)] /x
=lim(x->0) [1-(1-x+x²)] /{ x *[ 1+(1-x+x²)^(1/3)+(1-x+x²)^(2/3) ] }
=lim(x->0) [ x-x² ] /{ x *[ 1+(1-x+x²)^(1/3)+(1-x+x²)^(2/3) ] }
=lim(x->0) [ 1- x ] /[ 1+(1-x+x²)^(1/3)+(1-x+x²)^(2/3) ]
= 1/[1+1+1]
= 1/3