$f\left(n^2+1\right)=\frac{n^2+1}{1+2\cdot \left(n^2+1\right)}=\frac{n^2+1}{1+2n^2+2}=\frac{n^2+1}{2n^2+3}=\frac{n^2\left(1+\frac{1}{n^2}\right)}{n^2\left(2+\frac{3}{n^2}\right)}\stackrel{n\to \infty }{\to }\frac{1+0}{2+0}=\frac{1}{2}$