<s>[INST] Find limit of sequence $\lim_{n\to\infty} \frac{n}{\sqrt{n^2+n+1}} $ I need help to find the limit:
$$\lim_{n\to\infty} \frac{n}{\sqrt{n^2+n+1}} $$
I've tried this:
$$\lim_{n\to\infty} \frac{\sqrt{n^2}}{\sqrt{n^2+n+1}} =\lim_{n\to\infty} \sqrt{\frac{n^2}{n^2+n+1}} = \lim_{n\to\infty} \sqrt{\frac{1}{1+\frac{1}{n}+\frac{1}{n^2}}}$$
but what's next?
 [/INST] Since
$n^2
\lt n^2+n+1
\lt (n+1)^2
=n^2+2n+1
$,
$n
\lt \sqrt{n^2+n+1}
\lt n+1
$
so
$1
\gt \dfrac{n}{\sqrt{n^2+n+1}}
\gt \dfrac{n}{n+1}
= 1-\dfrac1{n+1}
$.
Therefore
$\lim_{n\to\infty} \dfrac{n}{\sqrt{n^2+n+1}}
=1
$.
</s>