Question

这道求极限的题目怎么解？

Answer 1

x→+∞
lim (x^3+3x)^(1/3) - (x^2-2x)^(1/2)
=lim (x^3+3x)^(1/3) - x + x - (x^2-2x)^(1/2)
=lim (x^3+3x)^(1/3) - (x^3)^(1/3) + (x^2)^(1/2) - (x^2-2x)^(1/2)
记，a=(x^3+3x)^(1/3)；b=x=(x^3)^(1/3)
=lim (a-b)(a^2+ab+b^2)/(a^2+ab+b^2) + (x^2)^(1/2) - (x^2-2x)^(1/2)
=lim (a^3-b^3)/(a^2+ab+b^2)
+ [(x^2)^(1/2) - (x^2-2x)^(1/2)][(x^2)^(1/2) + (x^2-2x)^(1/2)] / [(x^2)^(1/2) + (x^2-2x)^(1/2)]
=lim (3x)/(a^2+ab+b^2) + (2x)/[(x^2)^(1/2) + (x^2-2x)^(1/2)]
=lim (3x)/(a^2+ab+b^2)+(2x)/(x+(x^2-2x)^(1/2))
=0+2/(1+1)
=1
有不懂欢迎追问

Answer 2

• 两个根式同时除以x乘以x（乘在根式外，除在根式里），把无界变有界，把无穷大变无穷小

• 无穷大与有界的0的积为0

Answer 3