Solving ϵ-N Proofs: Simplifying the Denominator with sqrt(2)

[Oshada]

ϵ-N proof

Homework Statement

Homework Equations

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.

 

[micromass]

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.

 

[Oshada]

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!

 

[micromass]

Should I prove that by induction?

Yes.

Also, once I've resolved the denominator, how should I go about with the denominator?

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.

And any help with the limit theorem explanation would be very handy. Thanks!

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.

 

[Oshada]

Thank you very much! Does the limit go to 0? I got 0 from both N and limit theorems (N was sqrt(20/ϵ))

 

[micromass]

Yes, the limit will go to 0!