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Math Help - Limit = e^x

  1. #1
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    Limit = e^x

    Not even sure how to approach this one:

    (I have to write this without latex, since it seems to be down right now)

    Show that lim (n-->infinity) (1+x/n)^n = e^x

    My only thought is to restate this in a limit definition fashion, but it's not getting me anywhere:

    lim (n-->infinity) n ln(1+x/n)

    I know that lim (n-->infinity) (1+1/n)^n = e, but I can't seem to find the connection between these two equations.

    Any help is appreciated.

    Thanks!
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  2. #2
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    Are you allowed to use L'Hospital's Rule? Are you allowed to use the facts that and ?
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  3. #3
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    Not L'Hospitals rule quite yet. That's still a few sections away. We have the other two, though.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Express:

    ( 1 + x/n )^n = [ ( 1 +1/(n/x) )^ ( n/x ) ]^x
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    Thank you, but I have no idea how to explain how this is mathematically valid. I have no doubt that it is. It's just way over my head.
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  6. #6
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    Quote Originally Posted by joatmon View Post
    Thank you, but I have no idea how to explain how this is mathematically valid. I have no doubt that it is. It's just way over my head.
    You should know that .

    You should also note that the stuff in the brackets is of the form , which is a VERY well-known limit...
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  7. #7
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by joatmon View Post
    Not even sure how to approach this one:

    (I have to write this without latex, since it seems to be down right now)

    Show that lim (n-->infinity) (1+x/n)^n = e^x

    My only thought is to restate this in a limit definition fashion, but it's not getting me anywhere:

    lim (n-->infinity) n ln(1+x/n)

    I know that lim (n-->infinity) (1+1/n)^n = e, but I can't seem to find the connection between these two equations.

    Any help is appreciated.

    Thanks!
    With standard 'binomial expansions'...

    (1)

    ... and a little of 'patience' You can demonstrate that is...

    (2)

    ... and the second limit is just e^{x} ...

    Kind regards

    \chi \sigma
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