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When is it OK to pass the limit to the exponent

  1. Sep 22, 2009 #1
    when is it justified to do something like this

    [tex] \lim_{n\to\infty} e^{lnx^{1/n}} = e^{\displaystyle\lim_{n\to\infty}lnx^{1/n}}[/tex]

    or something like this

    [tex] \lim_{n\to\infty} 2^{f(n)} = 2^{\displaystyle\lim_{n\to\infty}f(n)}}[/tex] ?

    I am assuming that I can do something like this in both cases, but why?

    thank you.
     
  2. jcsd
  3. Sep 22, 2009 #2

    lurflurf

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    Homework Helper

    in general
    lim f(g(n))=f(lim g(n))
    holds when f is continuous at lim g(n)
    exponential functions are everywhere continuous so this can be done in both cases
     
  4. Sep 22, 2009 #3

    HallsofIvy

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    In fact [itex]\lim_{x\to a}f(x)= f(\lim_{x\to a} x)= f(a)[/itex], from which [itex]\lim_{x\to a}f(g(x))= f(\lim_{x\to a}g(x))[/itex]
    is the definition of "f is continuous at x= a".
     
  5. Sep 23, 2009 #4
    Wherever it is continuous which is everywhere
     
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