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Thread: 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 "$\displaystyle \lim_{x\to a} f(x)= 0$" 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 "$\displaystyle \lim_{x\to a} f(x)= 0$" 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 $\displaystyle (a-\delta,a+\delta)$ on which $\displaystyle |f(x)|<0.5~\&~g(x)>1$.

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

    Now as $\displaystyle \delta\to 0^+$, 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 $\displaystyle g(x)\to\infty$ then $\displaystyle (0.5)^{g(x)}\to 0$.
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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 $\displaystyle g(x)\to\infty$ then $\displaystyle (0.5)^{g(x)}\to 0$.
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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 $\displaystyle h(x)\le f(x)\le g(x)$ and $\displaystyle \lim _{x \to a} h(x) = \lim _{x \to a} g(x)=L$
    then $\displaystyle \lim _{x \to a} f(x) = L$
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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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    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 $\displaystyle |r|<1$ then $\displaystyle \lim _{x \to \infty } r^x = 0$.

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

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

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

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