PROBLEM 3. Calculate $\lim _{n \rightarrow \infty}\left\{\sqrt{n^{2}+5 n+9}\right\}$, where $\{x\}$ represents the fractional part of the real number $x$.
## PROBLEM 3.

(1p) We have $\left\{\sqrt{n^{2}+5 n+9}\right\}=\sqrt{n^{2}+5 n+9}-\left[\sqrt{n^{2}+5 n+9}\right]$

(3p) Since, $n+2<\sqrt{n^{2}+5 n+9}<n+3, \forall n \geqslant 1$

(1p) we obtain that $\left[\sqrt{n^{2}+5 n+9}\right]=n+2$

(2p) and thus, $\lim _{n \rightarrow \infty}\left\{\sqrt{n^{2}+5 n+9}\right\}=\lim _{n \rightarrow \infty}\left(\sqrt{n^{2}+5 n+9}-(n+2)\right)=\frac{1}{2}$