JEE Math Practice Question

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0)=0, f\left(\frac{\pi}{2}\right)=3$ and $f^{\prime}(0)=1$. If \[ g(x)=\int_{x}^{\frac{\pi}{2}}\left[f^{\prime}(t) \operatorname{cosec} t-\cot t \operatorname{cosec} t f(t)\right] d t \] for $x \in\left(0, \frac{\pi}{2}\right]$, then what is the $\lim _{x \rightarrow 0} g(x)$?