Homework Help: -N proof

1. Aug 24, 2011

ϵ-N proof

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

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. Aug 24, 2011

micromass

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

$$n^3\leq n^3+2n-\sqrt{2}$$

The $\sqrt{2}$ is just a red herring. It's just a constant.

3. Aug 24, 2011

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!

4. Aug 24, 2011

micromass

Re: ϵ-N proof

Yes.

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

$$\left|\frac{7n+13}{n^3+2n-\sqrt{2}}\right|\leq \left|\frac{7n+13}{n^3}\right|$$

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.

5. Aug 25, 2011

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))

6. Aug 25, 2011

micromass

Re: ϵ-N proof

Yes, the limit will go to 0!!