Find the supremum of the following set:

Question

Find the supremum of the following set and justify why: $$ A:= \left\{\frac{n^{2}+6}{n+8} \ \Bigg|\ n\in\mathbb{N}\right\} . $$

Answer 1

Note that $\frac{n^{2}+6}{n+8}\to\infty$ as $n\to\infty$.

Answer 2

Is there any upper bound to $\frac{n^2+6}{n+8}$? Can you find any number that is larger than $\frac{n^2+6}{n+8}$ for all $n$?

What if instead of $\frac{n^2+6}{n+8}$ you just had $\frac{n^2}{n} = n$? Do the additional constants change the asymptotic behavior for large $n$? What does that tell you about the possibility of any finite supremum?