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Proving limit relations

  1. Aug 8, 2013 #1
    I have been asked to prove the following limit relations.
    (a) lim(as x goes to infinity) (b^x-1)/x = log(b)

    (b) lim log(1+x)/x = 1

    (c) lim (1+x)^(1/x) = e

    (d) lim (1+x/n)^n =e^x

    Unfortunately, I really have no idea where to start. We have a theorem that says if f(x)=the sum of (c sub n)*(x^n) then the limit of f(x) is the sum of c sub n. Is that useful for this problem? Any suggestions on how to do this?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 8, 2013 #2

    Zondrina

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    How rigorous is your class? Do you prove things using the ##\epsilon - \delta## definition or are you simply trying to show that the limits are what they are?
     
  4. Aug 8, 2013 #3
    For this particular problem, I think we are just supposed to show that they are what they are.
     
  5. Aug 8, 2013 #4

    Zondrina

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    Let's try the first one : ##lim_{x→∞} \frac{b^x-1}{x}##

    Try plugging in some values and see what happens for x = 1, 2, 3... . That should draw your attention to what is happening in the numerator depending on ##b##.
     
  6. Aug 8, 2013 #5

    HallsofIvy

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    The "standard" rules of limits are not sufficient here. But you should think about this: what definition of "e" are you using?
     
  7. Aug 8, 2013 #6

    vela

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    That can't be right. If b>1, the exponential function will grow much faster than ##x##, so the limit will diverge.
     
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