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Math Help - Find d such that f(x) converges

  1. #1
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    Find d such that f(x) converges

    EDIT: I see now to late that the title of the thread says: "Find d..." , but it is supposed to be: "Find c..."

    Let:
    f(x)=x-(4+x^4+x^6)^c

    The question is, what is c such that when x\to\infty f(x)\to k
    where k is some finite number.

    Now what I did is something like this:
    \lim\limits_{x\to\infty}x-(4+x^4+x^6)^c=\lim\limits_{x\to\infty}x-\lim\limits_{x\to\infty}(x^6(\frac{4}{x^6}+\frac{1  }{x^2}+1))^c
    =\lim\limits_{x\to\infty}x + \lim\limits_{x\to\infty}x^{6c} \cdot \lim\limits_{x\to\infty}(\frac{4}{x^6}+\frac{1}{x^  2}+1)^c
    =\lim\limits_{x\to\infty}x-\lim\limits_{x\to\infty}x^{6c}=\lim\limits_{x\to\i  nfty}x-x^{6c}=k where k\in\mathbb{R}

    Now the only way such a limit is to converge to k is that x have equal powers, that is
    c=\frac{1}{6}, then k=0.

    Now the main problem is: How do I prove there is no other c that such that f(x) converges when x\to\infty ? I just need you to point me in the right direction.
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  2. #2
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    Nothing ?

    Any way, I think I am supposed to express (4+x^4+x^6)^c as a taylor-series.
    I just started to learn Taylor series today, so I have no idea how I am supposed to do that.
    Can someone please clarify?
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