Let $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ be defined as

$\mathrm{f}\left(\mathrm{x}\right)=$$\left\{\begin{array}{ll}\underset{\mathrm{n}\to \infty }{\lim }\left(\frac{\left[\mathrm{x}\right]}{1+{\mathrm{n}}^2}+\frac{3\left[\mathrm{x}\right]}{2+{\mathrm{n}}^2}+\frac{5\left[\mathrm{x}\right]}{3+{\mathrm{n}}^2}+.....+\frac{\left(2\mathrm{n}-1\right)\left[\mathrm{x}\right]}{\mathrm{n}+{\mathrm{n}}^2}\right)& ; \mathrm{x}\ne \frac{\mathrm{\pi }}{2}\\ 1& \mathrm{when} \mathrm{x}=\frac{\mathrm{\pi }}{2}\end{array}\right.$(Where [.] denotes greatest integer function), then which of following statement (s) is/are correct?