QJEEM11Ma4376: JEE Main 2011 Mathematics - Class 12 JEEM11Ma4376 Solution: The value of and for which the function f ( x ) = { sin ⁡ ( p + …

Question

Verified Solution • 2025–26 Session moderate

The value of

p

and

q

for which the function

f (x) = {\begin{matrix} \frac{\sin (p + 1) x + \sin x}{x} & , x < 0 \\ q & , x = 0 \\ \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}} & , x > 0 \end{matrix}

is continuous for all

x

in R, are

Accepted Answer

L.H.L. = \mathop {\lim }\limits_{x \to 0} f\left( x \right) = \mathop {\lim }\limits_{h \to 0} {{\sin \left\{ {\left( {p + 1} \right)\left( { - h} \right)} \right\} - \sin \left( { - h} \right)} \over { - h}}

= \mathop {\lim }\limits_{h \to 0} {{ - \sin \left( {p + 1} \right)h} \over { - h}} + {{\sin \left( { - h} \right)} \over { - h}} = p + 1 + 1 = p + 2

R.H.L.

= \mathop {\lim }\limits_{x \to {\sigma ^ + }} f\left( x \right) = \mathop {\lim }\limits_{h \to 0} {{\sqrt {1 + h} - 1} \over h}

= \mathop {\lim }\limits_{h \to 0} {1 \over {\left( {\sqrt {1 + h} + 1} \right)}} = {1 \over 2}

and

f\left( 0 \right) = q \Rightarrow p = - {3 \over 2},q = {1 \over 2}

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