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Math Help - Limit Evaluation 3rd root

  1. #1
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    Limit Evaluation 3rd root

    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.
    Attached Thumbnails Attached Thumbnails Limit Evaluation 3rd root-limit.bmp  
    Last edited by e2718281; October 23rd 2010 at 06:07 AM.
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  2. #2
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    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}
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  3. #3
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    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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