求高数题目.求极限,x+根号3（1-x^3),x趋向于正无穷...我看了解答.但是没看懂.希望能仔细讲解!

Question

求高数题目.求极限,x+根号3（1-x^3),x趋向于正无穷
...我看了解答.但是没看懂.希望能仔细讲解!

Answer 1

lim [x+(1-x³)^(1/3)]
分子分母同除以x
=lim [1+(1/x³ - 1)^(1/3)] / (1/x)
=lim [1-(1- 1/x³)^(1/3)] / (1/x)
=lim -[(1- 1/x³)^(1/3) - 1] / (1/x)
(1- 1/x³)^(1/3) - 1与-(1/3)(1/x³)是等价无穷小,因为：(1+x)^a-1等价于ax
=lim (1/3)(1/x³) / (1/x)
=lim (1/3)(1/x²)
=0
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