Let f   :   R → R be a differentiable function such that f 0 = 0 ,   f π 2 = 3 and…

$g\left(x\right)= {\int }_x^{\frac{\mathrm{\pi }}{2}}[f^{\text{'}}\left(t\right)cosec t-f\left(t\right)cosec t\text{ cot} t]dt$

$={\left[f\left(t\right)\cos ect\right]}_x^{\frac{\mathrm{\pi }}{2}}$

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

$\therefore \underset{x\to 0}{\lim }g\left(x\right)=3-\underset{x\to 0}{\lim }\frac{f\left(x\right)}{\sin x}=\underset{x\to 0}{\lim }\frac{f^{\text{'}}\left(x\right)}{\cos x}; as f^{\text{'}}\left(0\right)=1$

$\Rightarrow \underset{x\to 0}{\lim }g\left(x\right)=3-1=2$