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Having trouble solving this limit

  1. Sep 4, 2010 #1
    1. The problem statement, all variables and given/known data

    lim (1+(a/x))^(bx) as x-->infinity

    2. Relevant equations



    3. The attempt at a solution

    so, i raised the limit to e, and said e^lim(as x->inf) bxlog((a/x)+1). Then I pulled the constant b out and put it outside of the lim... I don't know how to do the rest though :( Was doing all my lim problems just fine until i came to this one
     
  2. jcsd
  3. Sep 4, 2010 #2

    Mark44

    Staff: Mentor

    Raising things as a power of e is the wrong direction to go. Instead, let y = (1 + a/x)^(bx).

    Now take the natural log of both sides, and then take the limit. You should get something you can use L'Hopital's rule on.
     
  4. Sep 4, 2010 #3

    Mentallic

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    Or, from where you left off, let y=a/x. Remember that this changes your limit from x approaches infinite to y approaches 0.
     
  5. Sep 4, 2010 #4

    Mentallic

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    Sorry, a rescaling would be much better. Let x=an
     
    Last edited: Sep 5, 2010
  6. Sep 4, 2010 #5

    Hurkyl

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    That's more or less exactly what he did, he just organized the work differently.
     
  7. Sep 5, 2010 #6

    hunt_mat

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    Do you know the power series expansion for log (1+x)? This will help you solve your problem.
     
  8. Sep 5, 2010 #7

    Mentallic

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    I don't think that's really necessary. It would be assumed obvious to let [tex]\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^{x}=e[/tex] so all that is required is to transform the question into a form that leaves this as the limit.
     
  9. Sep 5, 2010 #8

    hunt_mat

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    I'd be uneasy saying that as e is defined using the natural numbers not the real numbers
     
  10. Sep 5, 2010 #9

    Mentallic

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    What does x approaching infinite have to do with either the natural or the real numbers?
     
  11. Sep 5, 2010 #10

    hunt_mat

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    It still looks dodgy to me, I don't think you can say that. The route with expanding the log function is by far the safer route to go down.
     
  12. Sep 5, 2010 #11
    Here you go, the equivalent real value definition:

    [tex]
    \lim_{x\to0}\left(1+x\right)^{\frac{1}{x}}=e
    [/tex]

    just don't use it all in one place! :)
     
  13. Sep 5, 2010 #12

    hunt_mat

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    You then have to show that this in indeed the same value as the normal definition. No, I am convinced that there is far more work this way than my way.
     
  14. Sep 5, 2010 #13
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