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Analysis Limit Question

  1. May 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Q(y)=a0+a1y+.........+amy^m is a polynomial of degree m and I need to show that:

    Lim{y->Inf} Q(y)/ey2=0





    2. Relevant equations



    3. The attempt at a solution

    It seems obvious but I can't seem to be able to prove it, and don't really know where to start, any help would be much appreciated.
     
  2. jcsd
  3. May 20, 2012 #2
    Could you prove it for the monomial ##a_m y^m##? Have you learned L'Hôpital's rule yet?
     
  4. May 20, 2012 #3
    Oh ok, didn't think of using l'Hopitals rule. So can you just say:

    Lim_{y->Inf} Q(y)/ey2 <=> lim_{y->inf} Q(m)(y)/((2y)m).ey2)

    <=> Lim_{y->inf} a(m).m!/((2y)m).ey2)=0

    Is that right?
     
  5. May 20, 2012 #4
    Sorry just realised the bottom part of the limit is wrong as after you've differentiated once your gonna have to use the product rule after that. Could you just use induction on the degree m then, and then use L'hopitals rule to prove it for n+1?
     
  6. May 20, 2012 #5
    Sure, you could use induction if you want to be very rigorous. I would probably just be lazy and say, "applying L'Hôpital's rule ##m## times...", but induction is a more formal proof. This sort of argument should work for a monomial ##a_m y^m## or for Q(y) itself. Just make sure the hypotheses of L'Hôpital's rule are satisfied: you need to have an indeterminate form of ##0/0## or ##\infty/\infty##.
     
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