The values of p and q for which the function 

$f\left(x\right)$ $=\left\{\begin{array}{cl}\frac{\sin (\mathrm{p}+1)\mathrm{x}+\sin \mathrm{x}}{\mathrm{x}}& \text{ if }\mathrm{x}<0\\ \mathrm{q}& \text{ if }\mathrm{x}=0\\ \frac{\sqrt{\mathrm{x}+{\mathrm{x}}^2}-\sqrt{\mathrm{x}}}{{\mathrm{x}}^{3/2}}& \text{ if }\mathrm{x}>0\end{array}\right.$  is continuous $\mathrm{\forall }\mathrm{x}\in \mathrm{R}$ are

LHL = RHL =$\mathrm{f}\left(0\right)\Rightarrow$$\underset{x\to 0^{-}}{\lim }\frac{\sin (\mathrm{p}+1)\mathrm{x}+\sin \mathrm{x}}{\mathrm{x}}=\underset{x\to 0^{+}}{\lim }\frac{\sqrt{\mathrm{x}+{\mathrm{x}}^2}-\sqrt{\mathrm{x}}}{{\mathrm{x}}^{3/2}}=q$

 $\mathrm{P}+1+1=\mathrm{q}=\underset{\mathrm{x}\to 0}{lim} \frac{\sqrt{\mathrm{x}}(\sqrt{1+\mathrm{x}}-1)}{\sqrt{\mathrm{x}}\mathrm{x}}$=$\underset{x\to 0}{\lim }\frac{\left(1+x\right)-1}{x\left(\sqrt{1+x}+1\right)}=\underset{x\to 0}{\lim }\frac{1}{\sqrt{1+x}+1}$
$$\begin{gathered} \Rightarrow \mathrm{P}+2=\mathrm{q}=1/2 \\ \Rightarrow \mathrm{P}+2=1/2\therefore \mathrm{P}=1/2-2=-3/2 \\ \Rightarrow p=\frac{-3}{2}, \mathrm{q}=\frac{1}{2} \end{gathered}$$