Question
The values of $$p$$ and $$q$$ for which the function \[f\left( x \right) = \left\{ \begin{array}{l}
\frac{{\sin \left( {p + 1} \right)x + \sin \,x}}{x},\,x < 0\\
q,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0\\
\frac{{\sqrt {x + {x^2}} - \sqrt x }}{{{x^{\frac{3}{2}}}}},\,\,\,\,\,\,\,\,\,x > 0
\end{array} \right.\] is continuous for all $$x$$ in $$R,$$ are-
A.
$$p = \frac{5}{2},\,\,q = \frac{1}{2}$$
B.
$$p = - \frac{3}{2},\,\,q = \frac{1}{2}$$
C.
$$p = \frac{1}{2},\,\,q = \frac{3}{2}$$
D.
$$p = \frac{1}{2},\,\,q = - \frac{3}{2}$$
Answer :
$$p = - \frac{3}{2},\,\,q = \frac{1}{2}$$
Solution :
$$\eqalign{
& {\bf{L}}{\bf{.H}}{\bf{.L}}{\bf{.}} = \mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right) \cr
& = \mathop {\lim }\limits_{h \to 0} \frac{{\sin \left\{ {\left( {p + 1} \right)\left( { - h} \right)} \right\} - \sin \left( { - h} \right)}}{{ - h}} \cr
& = \mathop {\lim }\limits_{h \to 0} \frac{{ - \sin \left( {p + 1} \right)h}}{{ - h}} + \frac{{\sin \left( { - h} \right)}}{{ - h}} \cr
& = p + 1 + 1 \cr
& = p + 2 \cr
& {\bf{R}}{\bf{.H}}{\bf{.L}} = \mathop {\lim }\limits_{x \to {\sigma ^ + }} f\left( x \right) \cr
& = \mathop {\lim }\limits_{h \to 0} \frac{{\sqrt {1 + h} - 1}}{h} \cr
& = \mathop {\lim }\limits_{h \to 0} \frac{1}{{\left( {\sqrt {1 + h} + 1} \right)}} \cr
& = \frac{1}{2} \cr
& {\text{and }}f\left( 0 \right) = q\,\,\,\,\, \Rightarrow p = - \frac{3}{2},\,\,q = \frac{1}{2} \cr} $$