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?

That's nice, now observe that


*

*$\frac1n \to 0$

*$\frac1{n^2} \to 0$
therefore
$$\sqrt{\frac{1}{1+\frac{1}{n}+\frac{1}{n^2}}}\to \sqrt{\frac{1}{1+0+0}}$$
