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Limit proof?

  1. Oct 29, 2007 #1
    limit proof??

    well what i am trying to understand,actually proof is if we can get with the limit inside a power (exponent) if the exponent is irrational.
    Say we have any sequence (a_n) or any function f(x), let p be irrational then can we do the following, if yes why, if not why???

    1. for the sequence

    l
    [tex]\lim_{\substack{\\n\rightarrow \infty}} (a_n)^{p} =(\lim_{\substack{\\n\rightarrow \infty}} a_n)^{p}[/tex] ?????
    and
    2.[tex]\lim_{\substack{\\x\rightarrow x_o}} (f(x))^{p} =(\lim_{\substack{\\x\rightarrow x_o}} f(x))^{p} [/tex]

    I know how to prove this but only when p is from naturals. HOwever i have never come accross any such a problem. Only today suddenly this idea crossed my mind, so i thought i might get some suggesstions here.
    So what is the proper answer to this??


    thnx in advance

    P.S if you could tell me where i could find a proof for this, i would be really grateful.
     
  2. jcsd
  3. Oct 29, 2007 #2

    Zurtex

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    (Sorry, this post isn't planning to add much, I'm just rambling a little to see what I come up with).

    So for natrual numbers it's pretty simple assuming that:

    [tex] \lim_{n \rightarrow \infty} a_na_n = \left( \lim_{n \rightarrow \infty} a_n \right) \left( \lim_{n \rightarrow \infty} a_n \right)[/tex]

    First of all, I want to point out that this may not be true, take the sequence an = (-1)n

    So do you have to make a simmilar assumption with p as an irrational? I think so, I also think you have to clearly define what you're talk about when you mean anp where p is irrational, it's probabily wise to go back to that definition and try and build it up from there, clearly stating any assumptions.
     
  4. Oct 30, 2007 #3
    I am assuming that the limit of the sequence (a_n) actually exists, when p is natural, but what for example when p is irrational, this is what i am trying to show. let say a_n=(2n-1)/(n+1), so what can we say now for the limit

    [tex]\lim_{\substack{\\n\rightarrow \infty}} (a_n)^{p} =(\lim_{\substack{\\n\rightarrow \infty}} a_n)^{p}[/tex]

    when p is natural i can clearly go like this, as u stated

    [tex]\lim_{\substack{\\n\rightarrow \infty}} (a_n)^{p} =\lim_{n \rightarrow \infty} a_na_na_n ....a_n = \left( \lim_{n \rightarrow \infty} a_n \right) \left( \lim_{n \rightarrow \infty} a_n \right)\left( \lim_{n \rightarrow \infty} a_n \right)\left( \lim_{n \rightarrow \infty} a_n \right)......\left( \lim_{n \rightarrow \infty} a_n \right)=(\lim_{\substack{\\n\rightarrow \infty}} a_n)^{p}[/tex]

    But again i cannot figure out how to do it when we have p irrational?? I can clearly not performe the same thing as i did above assuming that p natural.
     
    Last edited: Oct 30, 2007
  5. Oct 30, 2007 #4

    matt grime

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    Follow Zurtex's advice: what is x^p for p irrational? It is exp{p logx}, and log is continuous, multiplication by p is continuous, and exp is continuous.
     
  6. Oct 30, 2007 #5
    ohoho:surprised

    I think i need to be much more vigilent in the future. I think i got it now.

    Many thanks to u guys.

    P.s. If by any chance,in the future, i might encounter any other problems, or need further clarifications, concerning these kind of problems, i will be back. I hope u won't mind.
     
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