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-N proof

  1. Aug 24, 2011 #1
    ϵ-N proof

    1. The problem statement, all variables and given/known data

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    2. Relevant equations



    3. The attempt at a solution

    I've tried to make the denominator smaller as is usual with ϵ-N proofs. But the sqrt(2) confuses me. Any help is much appreciated.
     
  2. jcsd
  3. Aug 24, 2011 #2

    micromass

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    Re: ϵ-N proof

    Indeed, try to make the denominator smaller. Try to prove that there is an N such that for all n>N

    [tex]n^3\leq n^3+2n-\sqrt{2}[/tex]

    The [itex]\sqrt{2}[/itex] is just a red herring. It's just a constant.
     
  4. Aug 24, 2011 #3
    Re: ϵ-N proof

    Should I prove that by induction? Also, once I've resolved the denominator, how should I go about with the denominator? And any help with the limit theorem explanation would be very handy. Thanks!
     
  5. Aug 24, 2011 #4

    micromass

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    Re: ϵ-N proof

    Yes.

    If you're done with the denominator, then you have

    [tex]\left|\frac{7n+13}{n^3+2n-\sqrt{2}}\right|\leq \left|\frac{7n+13}{n^3}\right|[/tex]

    You may want to eliminate the constant 13 by making the numerator bigger.

    Well, the trick is basically to bring n in front of the numerator and to bring n3 in front of the denominator. Then you can eliminate an n and you can calculate the limit using the limit rules.
     
  6. Aug 25, 2011 #5
    Re: ϵ-N proof

    Thank you very much! Does the limit go to 0? I got 0 from both N and limit theorems (N was sqrt(20/ϵ))
     
  7. Aug 25, 2011 #6

    micromass

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    Re: ϵ-N proof

    Yes, the limit will go to 0!!
     
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