\lim_{n\to \infty } \frac{\, n^2+n+3-n^2}{\sqrt{n^2+n+3}+n}=\lim_{n\to \infty } \frac{\, n+3}{\sqrt{n^2\left(1+\frac{1}{n}+\frac{3}{n^2}\right)}+n}=\lim_{n\to \infty } \frac{\, n+3}{n\sqrt{\left(1+\frac{1}{n}+\frac{3}{n^2}\right)}+n}=\lim_{n\to \infty } \frac{\, n\left(1+\frac{3}{n}\right)}{n\left(\sqrt{\left(1+\frac{1}{n}+\frac{3}{n^2}\right)}+1\right)}=\frac{1+\frac{3}{\infty }}{\sqrt{1+\frac{1}{\infty }+\frac{3}{\infty ^2}}+1}=\frac{1}{1+1}=\frac{1}{2}.