Let $f:\mathit{R}\to \mathit{R}$ be a differentiable function such that $f\left(\frac{\pi }{4}\right)=\sqrt{2},f\left(\frac{\pi }{2}\right)=0$ and $f^{\text{'}}\left(\frac{\pi }{2}\right)=1$ and let $g\left(x\right)={\int }_x^{\frac{\pi }{4}}\left(f^{\text{'}}\left(t\right)\sec t+\tan t\sec tf\left(t\right)\right)dt$ for $x\in \left[\frac{\pi }{4},\frac{\pi }{2}\right).$ Then $\underset{x\to {\left(\frac{\pi }{2}\right)}^{-}}{\lim }g\left(x\right)$ is equal to