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Quick limit question

  1. May 12, 2010 #1
    how do you calculate lim x-> inf 4^x/7^x+4

    It seems obvious that the bottom is growing at a faster rate than the top, so the limit would be zero, but how would you show that algebraically? I tried using l'hopital and just wind up with:

    4^x ln4 / 7^x ln7

    and it doesn't seem that using l'hopital again and again would get me anywhere further.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 12, 2010 #2
    Personally, I would devide top and bottom by 7^x.
     
  4. May 12, 2010 #3
    ok so that leaves lim x->inf (4/7)^x

    Is that enough proof to show it goes to zero?
     
  5. May 12, 2010 #4
    if |a|<1 then lim(x-->infinity) for a^x = 0
    If you see the graph for any function a^x where |a|<1, you see this is true
    You can prove that by using the convergence's tests for the infinite series ..
     
  6. May 12, 2010 #5
    ah ok,

    so that would be a geometric series with l r l < 1 (which we know converges), or you can prove this converges by using the root test, 4/7 < 1 , so since the series converges the limit of the sequence or partial sums goes to zero?

    do I have that all correct?
     
  7. May 12, 2010 #6
    sorry, I meant the limit of the nth partial sum goes to zero as n goes to infinity
     
  8. Jun 3, 2010 #7
    No.
    The limit of the nth term is zero.
    Not the limit of the sequence of the partial sums!
     
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