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Homework Help: Limit computation

  1. Jan 19, 2006 #1

    MathematicalPhysicist

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    i need to compute the lim of (1+2/x)^x or (1-1/x)^x and (1+1/x)^(x+3) when x approaches infinity.
    if you can provide a steady method to compute it, it will be appreciated.

    btw i know that lim (1+1/x)^x as x->inf is "e", but does it have any correlation to here.
     
  2. jcsd
  3. Jan 19, 2006 #2

    ehild

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    Yes, it does.
    For the first problem, replace x/2=y. You should determine lim (1+1/y)^(2y) as y->infinity.
    lim (1+1/y)^(2y)=lim ((1+1/y)^2)->
    (lim (1+1/y))^2 = e^2 if y->infinity. But this is equivalent with the original limit when x -> infinity.
    As for the second problem, (1+1/x)^(x+3)=((1+1/x)^x)*(1+1/x)^3. You can proceed from here.
    In case of the third problem
    1-1/x=1/[1+1/(x-1)]
    Let be y= x-1. If x -> infinity , so does y.
    You have to detemine the limit
    lim (1-1/x)^x when x->infinity. It is equivalent with
    lim[(1/(1+1/y)]^(y+1)) when y->infinity.
    lim[(1/(1+1/y)]^(y+1))=
    1/lim[(1+1/y)^(y+1)]=1/lim[(1+1/y)^y*(1+1/y)] =1/[lim(1+1/y)^y*lim(1+1/y)]=1/e
    ehild
     
  4. Jan 19, 2006 #3
    arrange and factorize
     
  5. Jan 22, 2006 #4

    MathematicalPhysicist

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    thanks, i think it's a simple substitution.
     
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