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解:在(1-三次根号(1-x+x²))/x 的分子分母
同时乘以 :1+三次根号(1-x+x²)+三次根号(1-x+x²)^2
分子用立方差公式后,可与分母约去x, 化简得:
(1-x)/(1+三次根号(1-x+x²)+三次根号(1-x+x²)^2 )
此时分子分母的极限都存在, 由除法极限法则即可求出
函数(1-三次根号(1-x+x²))/x的极限是 1/3
同时乘以 :1+三次根号(1-x+x²)+三次根号(1-x+x²)^2
分子用立方差公式后,可与分母约去x, 化简得:
(1-x)/(1+三次根号(1-x+x²)+三次根号(1-x+x²)^2 )
此时分子分母的极限都存在, 由除法极限法则即可求出
函数(1-三次根号(1-x+x²))/x的极限是 1/3
展开全部
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
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
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