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Math Help - prove 0 to the infinite power is not indeterminate

  1. #1
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    prove 0 to the infinite power is not indeterminate

    suppose f is a positive function. if limx->a f(x) = 0 and limx->a g(x) = infinity, show that

    limx->a [f(x)]g(x) = 0

    this shows that 0 to the infinite power is not an indeterminate form.
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  2. #2
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    Re: prove 0 to the infinite power is not indeterminate

    What does " \lim_{x\to a} f(x)= 0" mean? And what does "[tex]\lim_{x\to a} g(x)= \infty[/itex]" mean?
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    Re: prove 0 to the infinite power is not indeterminate

    this is where the problem came from prove 0 to the infinite power is not indeterminate-img_4226.jpg
    Quote Originally Posted by HallsofIvy View Post
    What does " \lim_{x\to a} f(x)= 0" mean? And what does "[tex]\lim_{x\to a} g(x)= \infty[/itex]" mean?
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    Re: prove 0 to the infinite power is not indeterminate

    here is a hint:

    show log(f(x))*g(x) is very large negative, for x sufficiently close to a.
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  5. #5
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    Re: prove 0 to the infinite power is not indeterminate

    Quote Originally Posted by pnfuller View Post
    suppose f is a positive function. if limx->a f(x) = 0 and limx->a g(x) = infinity, show that
    limx->a [f(x)]g(x) = 0
    this shows that 0 to the infinite power is not an indeterminate form.
    There is an open interval (a-\delta,a+\delta) on which |f(x)|<0.5~\&~g(x)>1.

    So there |f(x)|^{g(x)}<(0.5)^{g(x)}.

    Now as \delta\to 0^+, what does that tell you?
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    Re: prove 0 to the infinite power is not indeterminate

    but how can you do that if the log(f(x)) is undefined at 0?
    Quote Originally Posted by Deveno View Post
    here is a hint:

    show log(f(x))*g(x) is very large negative, for x sufficiently close to a.
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    Re: prove 0 to the infinite power is not indeterminate

    what do you mean?
    Quote Originally Posted by Deveno View Post
    here is a hint:

    show log(f(x))*g(x) is very large negative, for x sufficiently close to a.
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    Re: prove 0 to the infinite power is not indeterminate

    Quote Originally Posted by pnfuller View Post
    what do you mean?
    Look at reply #5.
    If g(x)\to\infty then (0.5)^{g(x)}\to 0.
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    Re: prove 0 to the infinite power is not indeterminate

    but i have to show that using a definition or something, my teacher says using numbers as examples to prove something really isnt a proof.
    Quote Originally Posted by Plato View Post
    Look at reply #5.
    If g(x)\to\infty then (0.5)^{g(x)}\to 0.
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    Re: prove 0 to the infinite power is not indeterminate

    Quote Originally Posted by pnfuller View Post
    but i have to show that using a definition or something, my teacher says using numbers as examples to prove something really isnt a proof.
    Tell your teacher that this is a well known principle.
    If h(x)\le f(x)\le g(x) and \lim _{x \to a} h(x) = \lim _{x \to a} g(x)=L
    then \lim _{x \to a} f(x) = L
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    Re: prove 0 to the infinite power is not indeterminate

    i can't read post 5 and the weird little symbols it has
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    Re: prove 0 to the infinite power is not indeterminate

    Quote Originally Posted by pnfuller View Post
    i can't read post 5 and the weird little symbols it has
    If that is true, I would say that the original question is over you head.
    Your teacher had no business asking you to do it.
    It is a higher level concept question.
    To answer it properly one needs higher level understanding of limits than you seen to have.
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    Re: prove 0 to the infinite power is not indeterminate

    its an extra credit problem and i need the extra credit so i was really trying to figure it out!
    Quote Originally Posted by Plato View Post
    If that is true, I would say that the original question is over you head.
    Your teacher had no business asking you to do it.
    It is a higher level concept question.
    To answer it properly one needs higher level understanding of limits than you seen to have.
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  14. #14
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    Re: prove 0 to the infinite power is not indeterminate

    Quote Originally Posted by pnfuller View Post
    its an extra credit problem and i need the extra credit so i was really trying to figure it out!
    If I were you, I would say it is a known fact that:
    if |r|<1 then \lim _{x \to \infty } r^x  = 0.

    Because \lim _{x \to a } f(x)  = 0 then 0\le |f(x)|<0.5 for all x near a.

    But we are given \lim _{x \to a } g(x)  = \infty so (0.5)^{g(x)}\to 0.

    So \lim _{x \to a} f(x)^{g(x)}  = 0 because 0\le|f(x)|^{g(x)}\le (0.5)^{g(x)}\to 0

    Note that |f(x)|^{g(x)}=|f(x)^{g(x)}|
    Thanks from pnfuller
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