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One raised to the infinity power help please

  1. Sep 26, 2008 #1
    one raised to the infinity power help please!!!

    Let a and b be positive real numbers. For real number p define, f(p) = ((a^p + b^p)/2)^(1/p). Evaluate the limit of f(p) as p approaches 0.

    By directly plugging in zero, you would get (1)^inf. Wouldn't that equal 1 or would it be something else? When I put it into my 89 i got 1 as my answer, but for some reason I don't think that it is right. Please help!
     
  2. jcsd
  3. Sep 26, 2008 #2
    Re: one raised to the infinity power help please!!!

    why wouldn't it? 1x1x1x1... is still 1 isn't it?

    you can also check the graph at 0 to see if it converges to 1 when you plug in positive real numbers for a and b. don't the 89s take limits?
     
  4. Sep 26, 2008 #3
    Re: one raised to the infinity power help please!!!

    [tex]1^{\infty}[/tex] is undefined!
     
  5. Sep 26, 2008 #4

    arildno

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    Re: one raised to the infinity power help please!!!

    If a=b, note that the limit is "a".

    Let then "a" be greater than "b", and rewrite:
    [tex](\frac{(a^{p}+b^{p}}{2})^{\frac{1}{p}}=\frac{a}{2^{\frac{1}{p}}}({1+\frac{1}{N})^{\frac{1}{p}}, \frac{1}{N}=(\frac{b}{a})^{p}[/tex]
    Then,
    [tex]\frac{1}{p}=N\frac{\ln(\frac{a}{b})}{N\ln(N)}[/tex]


    Therefore, you may rewrite this as:
    [tex]((1+\frac{1}{N})^{N})^{\frac{\ln(\frac{a}{b})}{N\ln(N}}}[/tex]
    which remains nasty..
     
    Last edited: Sep 26, 2008
  6. Sep 26, 2008 #5

    lurflurf

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    Re: one raised to the infinity power help please!!!

    "If a=b, note that the limit is "a"."
    Indeed.
    "Wouldn't that equal 1"
    No.
    That is what is called an indeterminite form.
    further analysis is needed.
    f(p)=((a^p + b^p)/2)^(1/p)
    Let us consider an approximation that is exact in the limit.
    a^p~1+p log(a)
    b^p~1+p log(b)
    if p~0
    thus
    (a^p + b^p)/2~1+p log sqrt(ab)
    if p~0
    (1+x)^(1/P)~exp(x/p)
    if p~0 and x~0

    by combining these the answer should be clear.
     
  7. Sep 26, 2008 #6
    Re: one raised to the infinity power help please!!!

    Hint: if your sequence is [tex]f(p)[/tex], consider the limit of the related sequence [tex]g(p)[/tex] where [tex]g=\log f[/tex]. Then use some standard properties of continuity.
     
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