Maximum and minimum of a set

[Pittul]

Hello, I hope I'm posting this in the right section (I'm Italian and the classes have different names here).
I don't understand how to find the minimum and maximum of a set. I thought that I had to do the limits to the smallest and the biggest number of the domain, but I don't think it's correct because I can't solve this kind of exercises:

A={[2n+(-1)ⁿ sqrt(n²​+1)]/n}

I have to determine if this set has a maximum and/or a minimum.

Thank you!

 

[pka]

Hello, I hope I'm posting this in the right section (I'm Italian and the classes have different names here). I don't understand how to find the minimum and maximum of a set. I thought that I had to do the limits to the smallest and the biggest number of the domain, but I don't think it's correct because I can't solve this kind of exercises:
A={[2n+(-1)ⁿ sqrt(n²​+1)]/n}
I have to determine if this set has a maximum and/or a minimum.

Even realizing that is a translation, I not clear on what it is asking.
HERE is a plot of the expression.
I appears to be bounded an a large subset of the domain. But not having been given a domain it is impossible to be sure. Can you clarify?

 

[Pittul]

You're right, I'm sorry. The domain is N

 

[pka]

You're right, I'm sorry. The domain is N

That means that \(\displaystyle n>0\) so that \(\displaystyle \dfrac{2n+(-1)^n\sqrt{n^2+1}}{n}=2+(-1)^n\sqrt{1+\frac{1}{n^2}}\) which is bounded.

 

[Pittul]

How can you say that it's bounded? Have you done the limit to +infinity?

 

[HallsofIvy]

One way to approach this problem is to look at the associated continuous function:
\(\displaystyle A(x)= (2x+ (-1)^x\sqrt{x^2+ 1})^x\) for x> 0.

Find possible max and min for that function. If they are not at an integer value of x, look at the closest integers.

 

[pka]

That means that \(\displaystyle n>0\) so that \(\displaystyle \dfrac{2n+(-1)^n\sqrt{n^2+1}}{n}=2+(-1)^n\sqrt{1+\frac{1}{n^2}}\) which is bounded.

How can you say that it's bounded? Have you done the limit to +infinity?

It is is trivial, just look at it: \(\displaystyle (\forall n>0)\left[0\le 2+(-1)^n\sqrt{1+\frac{1}{n^2}}\le 4\right]\)
Surely that is a bounded set?

 

[Pittul]

Ok, thank you all very much