Let f: ℝ → ℝ be a differentiable function such that f(0) = 0,...

$g( x ) = {∫}_{x}^{\dfrac{\pi}{2}}[ {f}^{'}( t )cosect - f( t )cosect\text{cot}t ]dt$

$= {\left[f ( t ) cosect\right]}_{x}^{\dfrac{\pi}{2}}$

$= f\left(\dfrac{\pi}{2}\right)cosec\left(\dfrac{\pi}{2}\right) - \dfrac{f\left(x\right)}{sin x} = 3 - \dfrac{f\left(x\right)}{sin x}$

$∴ \lim_{x \to 0} g\left(x\right) = 3 - \lim_{x \to 0} \dfrac{f\left(x\right)}{sin x} = \lim_{x \to 0} \dfrac{{f}^{'}\left(x\right)}{cos x}; as{f}^{'}\left(0\right) = 1$

$⇒ \lim_{x \to 0} g\left(x\right) = 3 - 1 = 2$