Question

求极限：lim(x趋向于无穷) (x+(1-x^3)^(1/3))=? 求过程！

Answer 1

以下写极限符号时省略x的条件哈
设a=x , b=(1-x^3)^(1/3) , 因(a+b)(a^2-ab+b^2)=a^3+b^3
lim (x+(1-x^3)^(1/3))=lim (a+b)
=lim (a^3+b+3)/(a^2-ab+b^2)
=lim 1/(a^2-ab+b^2)
=lim 1/x^2[1-(-b)+b^2]
=1/lim x^2[(-b-1)^2-b]
=1/lim x^2{[(x^3-1)^(1/3)-1]^2+(x^3-1)^(1/3)}
然后......分母一块都非负，一块平方递增，倒数就出结果咯。

Answer 2

x³+[(1-x^3)^(1/3)]³ = (x+(1-x^3)^(1/3))(x²-x(1-x^3)^(1/3)+(1-x^3)^(2/3))
lim(x→∞) (x+(1-x^3)^(1/3))=lim(x→∞) {x³+[(1-x^3)^(1/3)]³}/(x²-x(1-x^3)^(1/3)+(1-x^3)^(2/3))
=lim(x→∞) [x³+(1-x^3)]/(x²-x(1-x^3)^(1/3)+(1-x^3)^(2/3))
=lim(x→∞) 1/(x²-x(1-x^3)^(1/3)+(1-x^3)^(2/3))
=lim(x→∞) (1/x²) / (1-(1/x^3-1)^(1/3)+(1/x^3-1)^(2/3))
=0/(1+1+1)
=0