If $f(x)= \begin{cases}\frac{\sin (p+1)+\sin x}{x} & , x<0 \\ q & x=0 \\ \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^{3 / 2}} & , x>0\end{cases}$
is continuous at x = 0, then the ordered pair (p,q) is equal to :
Correct Option: , 3
$\mathrm{RHL}=\lim _{x \rightarrow 0^{+}} \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^{\frac{3}{2}}}=\lim _{x \rightarrow 0^{+}} \frac{\sqrt{1+x}-1}{x}=\frac{1}{2}$
$\mathrm{LHL}=\lim _{\mathrm{x} \rightarrow 0} \frac{\sin (\mathrm{p}+1) \mathrm{x}+\sin x}{\mathrm{x}}=(\mathrm{p}+1)+1=\mathrm{p}+2$
for continuity $\mathrm{LHL}=\mathrm{RHL}=\mathrm{f}(0)$
$\Rightarrow(\mathrm{p}, \mathrm{q})=\left(\frac{-3}{2}, \frac{1}{2}\right)$
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