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Math Help - Prove the following limits when c ∈ N: (Analysis help)

  1. #1
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    Prove the following limits when c ∈ N: (Analysis help)

    Prove that when c ∈ N:

    a)
    lim [(1+x)^(1/c) - 1]/x = 1/c
    x->0

    b)
    lim [(1+x)^r - 1]/x = r
    x->0

    ,where r = c/n


    My approach for a and b are pretty similar, i get stuck at the same point.

    For a, i multiplied the numerator and the denominator by [(1+x)^(1/c) + 1] to get rid of the radical on top. This left me with:

    Lim 1/[(1+x)^(1/c)+1]
    x->0

    This is where im stuck, if i find the limit of the numerator and divide it by the limit of the denominator, i dont get 1/c. So im clearly doing something wrong.

    As for part c, i pretty much use the same approach and get:

    Lim 1/[(1+x)^(r)+1]
    x->0

    which also doesnt lead to the answer.

    Please help.
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  2. #2
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    l’Hopital’s rule takes care of both of those in one step.
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  3. #3
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    Here is a way to do without L'H˘pital's rule:

    1. \displaystyle \lim_{x\to{0}}\frac{(x+1)^{\frac{1}{c}}-1}{x} = \lim_{x\to{0}}\frac{(x+1)^{\frac{1}{c}}-1}{(x+1)-1}

    Put (x+1)^{\frac{1}{c}} = y, and since c\in\mathbb{N}, we have:

    \displaystyle \lim_{y\to{1}}\frac{y-1}{y^c-1} = \lim_{y\to{1}}\frac{y-1}{(y-1)(y^{c-1}+y^{c-2}+\cdots +1)}

     \displaystyle  = \lim_{y\to{1}}\frac{1}{(y^{c-1}+y^{c-2}+\cdots +1)} = \frac{1}{c}.

    2. Do the same.
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