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Homework Help: Limits help

  1. Nov 14, 2007 #1
    1. The problem statement, all variables and given/known data

    Find the limit as n tends to infinity of xn = (n^2 + exp(n))^(1/n)

    2. Relevant equations

    maybe use ( 1 + c/n )^n tends to exp(c)

    3. The attempt at a solution

    I know that inside the barckets are both inceasing and the 1/n makes it decrease but how do i find out which is stronger and what the limit is?
     
  2. jcsd
  3. Nov 14, 2007 #2

    JasonRox

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    Try finding the limit of the ln of the function.

    Tip: ln(a^b) = b ln(a)
     
  4. Nov 14, 2007 #3
    i already tried that and i cant see how it helps, (1/n)ln(n^2 + exp(n)) has the same problem...
     
  5. Nov 14, 2007 #4

    Dick

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    Try l'Hopital's rule, if you know that.
     
  6. Nov 14, 2007 #5
    i have a feeling im not allowed to use it, is there another way?
     
  7. Nov 14, 2007 #6

    Dick

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    What you really need to know is that n^2/exp(n)->0 as n->infinity. There are a variety of ways to show that - try to think of one. Once you done that then ln(exp(n)+n^2)=ln(exp(n)*(1+n^2/exp(n))=ln(exp(n))+ln(1+n^2/exp(n)) etc.
     
  8. Nov 14, 2007 #7
    [tex](n^2+e^n)^{1/n} = e(1+n^2/e^n)^{1/n}[/tex] now it is trivial but the fact that [tex](1+x_n/n)^{1/n} \to e^x[/tex] if [tex]x_n\to x[/tex].
     
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