The value of p and q for which the function f(x)={sin(p+1)x+sinxx , x 0 is continuous for all x in R, is

limx→0-sin(p+1)x+sinxx=q=limx→0+x+x2−xx3/2 limx→0-(p+1)cos(p+1)x+cosx=q=12⇒p+1+1=12⇒p=−32; q=12

The value of p and q for which the function f(x)={sin(p+1)x+sinxx , x 0 is continuous for all x in R, is

limx→0-sin(p+1)x+sinxx=q=limx→0+x+x2−xx3/2 limx→0-(p+1)cos(p+1)x+cosx=q=12⇒p+1+1=12⇒p=−32; q=12

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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, is

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$p = \frac{1}{2}, q = - \frac{3}{2}$

$p = \frac{5}{2}, q = \frac{1}{2}$

$p = \frac{1}{2}, q = \frac{3}{2}$

$p = - \frac{3}{2}, q = \frac{1}{2}$

answer is B.

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Detailed Solution

$\lim_{x \to 0^{-}} \frac{\sin (p + 1) x + \sin x}{x} = q = \lim_{x \to 0^{+}} \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}}$

$\lim_{x \to 0^{-}} (p + 1) \cos (p + 1) x + \cos x = q = \frac{1}{2}$

$\Rightarrow p + 1 + 1 = \frac{1}{2} \Rightarrow p = - \frac{3}{2}; q = \frac{1}{2}$

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