If f ( x ) * 20 c , ( p + 1 )x + ,x x , , , & x < 0 q , , , , , ,… - Vidyadip

MCQ

If $f\left( x \right)\left\{ {\begin{array}{*{20}{c}}
{\frac{{\sin \,\left( {p + 1} \right)x + \sin \,x}}{x},\,\,}&{x < 0} \\
{q\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,}&{x = 0} \\
{\frac{{\sqrt {x + {x^2}} - \sqrt x }}{{x/2}},}&{x > 0}
\end{array}} \right.$ Is continuous at $x = 0$, then the ordered pair $(p, q)$ is equal to

A
$\left( { - \frac{3}{2}, - \frac{1}{2}} \right)$
B
$\left( {\frac{5}{2},\frac{1}{2}} \right)$
C
$\left( { - \frac{1}{2},\frac{3}{2}} \right)$
✓
$\left( { - \frac{3}{2}, \frac{1}{2}} \right)$

✓

Answer

Correct option: D.

$\left( { - \frac{3}{2}, \frac{1}{2}} \right)$

d
$RHL = \mathop {\lim }\limits_{x \to {0^ + }} \frac{{\sqrt {x + {x^2}} - \sqrt x }}{{{x^{3/2}}}}$

$ = \mathop {\lim }\limits_{x \to {0^ + }} \frac{{\sqrt {1 + x} - 1}}{x} = \frac{1}{2}$

$LHL = \mathop {\lim }\limits_{x \to {0^ - }} \frac{{\sin \left( {\beta + 1} \right)x + \sin x}}{x}$

$ = \left( {p + 1} \right) + 1$

$ = p + 2$

For function to be continuous

$LHL = RHL = f\left( 0 \right)$

$ \Rightarrow \left( {p,q} \right) = \left( {\frac{{ - 3}}{2},\frac{1}{2}} \right)$

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