• # question_answer Iff(x)=\left\{ \begin{align} & \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{align} \right. is continuous at x = 0, then the ordered pair (p,q) is equal to :                         [JEE Main 10-4-2019 Morning] A) $\left( \frac{5}{2},\frac{1}{2} \right)$          B)   $\left( -\frac{3}{2},-\frac{1}{2} \right)$C) $\left( -\frac{1}{2},\frac{3}{2} \right)$                    D) $\left( -\frac{3}{2},\frac{1}{2} \right)$

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