Let

\[f(t)=\int_{0}^{\infty} \arctan \left(\frac{t}{\sinh x}ight) d x, \quad t>0\]

where \(\sinh x=\frac{1}{2}\left(e^{x}-e^{-x}ight)\).

(i) Show that \(f\) is differentiable on \((0, \infty)\), but \(f^{\prime}(0+)\) does not exist.

(ii) Find closed expressions for \(f^{\prime}, f(0)\) and \(\lim _{t ightarrow \infty} f(t)\).