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Math Help - Help Solving the Limit of this equation (third root)

  1. #1
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    Help Solving the Limit of this equation (third root)

    Find the solution for: [(1+cx)^1/3 - 1]/x as x approaches 0.

    Hello, I've been trying to solve this problem for a little over a month without much progress..

    Looking at it, the logical answer seems to be to get the (1+cx) out of it's third root form to cancel the constants and remove x, but I can't seem to do that w/o adding other similar terms. I've tried adding elements to the top to get a factorial out but that hasn't worked, I've tried multiplying by the positive counterpart ((1+cx)^1/3 + 1) to up the polynomial without luck, and lately I've been playing with natural logs to bring the 1/3 down; all to no avail.

    I know the limit exists for sure, any help would be appreciated.

    Thanks,
    Rhek
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  2. #2
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    Re: Help Solving the Limit of this equation (third root)

    Have you tried L'hospital's rule?

    L'H˘pital's rule - Wikipedia, the free encyclopedia
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  3. #3
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    Re: Help Solving the Limit of this equation (third root)

    I've never like using "L'Hopital" if a limit could be done without Calculus.

    Recall that x^3- y^3= (x- y)(x^2+ xy+ y^2)

    Taking y= 1,
    x^3- 1= (x- 1)(x^2+ x+ 1)

    Replacing x by a^{1/3}, x^3= a and x^2= a^{2/3} and a- 1= (a^{1/3}- 1)(a^{2/3}+ a^{1/3}+ 1)

    That is, ((1+cx)^{1/3}- 1)/x can be written as
    (1+cx- 1)/[x((1+cx)^{2/3}+ (1+ cx)^{1/3}+ 1)]= cx/[x((1+cx)^{2/3}+ (1+ cx)^{1/3}+ 1)] = c/((1+cx)^{2/3}+ (1+ cx)^{1/3}+ 1)

    And, the limit of that, as x goes to 0, is easy.
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    Re: Help Solving the Limit of this equation (third root)

    Thanks for both the responses guys. I should have stipulated that the problem is meant to be done w/o the use of derivatives, but now I/ve got both solutions. Thanks so much for the help, been scratching my head over that one for a long time.

    -Rhek
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