Limit Evaluation 3rd root

**e2718281** · October 23rd 2010, 05:02 AM

evaluate:

$\displaystyle\lim_{x \to \infty} (x+1)^{2/3}-(x-1)^{2/3}$

Please help me. I've been struggling with this problem for a long time.

**HallsofIvy** · October 23rd 2010, 07:15 AM

$a^3- b^3= (a- b)(a^2+ ab+ b^2)$

With $a= (x+1)^{2/3}$ and $b= (x- 1)^{2/3}$ ,

that says the $(x+1)^2- (x-1)^2= ((x+1)^{2/3}- (x-1)^{2/3})((x+1)^{4/3}+ ((x+1)(x-1))^{2/3}+ (x-1)^2)$ .

$(x+1)^{2/3}- (x-1)^{2/3}= \frac{(x+1)^2- (x-1)^2}{(x+1)^{4/3}+ ((x+1)(x-1))^{2/3}+ (x-1)^2}$

**Unbeatable0** · October 23rd 2010, 10:45 AM

Another way is squeeze, using the following inequality which is valid for all $x\ge\frac{1+\sqrt{5}}{2}$ :

$(x+1)^\frac{2}{3} \le (x-1)^\frac{2}{3}+(x-1)^{-\frac{1}{3}}$

(the reverse inequality is true for all $x\le\frac{1-\sqrt{5}}{2}$ , with which you can calculate the limit at $-\infty$ )

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