$a_n= \sqrt{n^2-1}-n$ superior and inferior limits

Question

I'm given the succession $$a_n= \sqrt{n^2-1}-n$$ and I should find the superior limit $ \limsup_{n \to \infty}a_n$ and the inferior limit $ \liminf_{n \to \infty}a_n$

This succession is defined $\forall n \ge 1$ , $n \in N$

if $n=1, a_1=-1$;

if $n=2, a_2=\sqrt{3}-2$;

if $n=3, a_3=\sqrt{8}-3$;

if $n=4, a_3=\sqrt{15}-4$

....

$\sqrt{n^2-1}< n \rightarrow \sqrt{n^2-1}- n <0 \rightarrow a_n<0, \forall n \ge1$

$ \lim_{n \to \infty}a_n = 0$

and considering the real function , its derivative is strictily positive.

In my opinion it should be $ \limsup_{n \to \infty}a_n=0$

$ \liminf_{n \to \infty}a_n=-1$

Is it right?

Answer 1

Since$$\lim_{n\to\infty}\sqrt{n^2+1}-n=\lim_{n\to\infty}\frac1{\sqrt{n^2+1}+n}=0,$$you have$$\limsup_{n\to\infty}\sqrt{n^2+1}-n=\liminf_{n\to\infty}\sqrt{n^2+1}-n=0.$$