evaluate:
$\displaystyle \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.
$\displaystyle a^3- b^3= (a- b)(a^2+ ab+ b^2)$
With $\displaystyle a= (x+1)^{2/3}$ and $\displaystyle b= (x- 1)^{2/3}$,
that says the $\displaystyle (x+1)^2- (x-1)^2= ((x+1)^{2/3}- (x-1)^{2/3})((x+1)^{4/3}+ ((x+1)(x-1))^{2/3}+ (x-1)^2)$.
$\displaystyle (x+1)^{2/3}- (x-1)^{2/3}= \frac{(x+1)^2- (x-1)^2}{(x+1)^{4/3}+ ((x+1)(x-1))^{2/3}+ (x-1)^2}$
Another way is squeeze, using the following inequality which is valid for all $\displaystyle x\ge\frac{1+\sqrt{5}}{2}$:
$\displaystyle (x+1)^\frac{2}{3} \le (x-1)^\frac{2}{3}+(x-1)^{-\frac{1}{3}}$
(the reverse inequality is true for all $\displaystyle x\le\frac{1-\sqrt{5}}{2}$, with which you can calculate the limit at $\displaystyle -\infty$)