高等数学证明题 根据函数极限的定义证明lim（x→a）x^(1/3)=a^(1/3)
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首先,a = 0 的情形易证,略.
对 a ≠ 0,可取 x 与 a 同号,注意到
|x^(2/3) + [x^(1/3)][a^(1/3)] + a^(2/3)|
>= |x^(2/3) - 2[x^(1/3)][a^(1/3)] + a^(2/3)|
= |x^(1/3) - a^(1/3)|^2,
可得
|x^(1/3) - a^(1/3)|
= |x - a|/|x^(2/3) + [x^(1/3)][a^(1/3)] + a^(2/3)|
对 a ≠ 0,可取 x 与 a 同号,注意到
|x^(2/3) + [x^(1/3)][a^(1/3)] + a^(2/3)|
>= |x^(2/3) - 2[x^(1/3)][a^(1/3)] + a^(2/3)|
= |x^(1/3) - a^(1/3)|^2,
可得
|x^(1/3) - a^(1/3)|
= |x - a|/|x^(2/3) + [x^(1/3)][a^(1/3)] + a^(2/3)|
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