The value of p and q for which the function f(x)= sin(p+1)x+sinxx; x 0 is continuous for all real values of x are

at x=0,f(0)=q for x 0, limx→0+f(x)=(h+h2)1/2−h1/2h3/2=12∴f is continuous ⇒ 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 real values of x are

at x=0,f(0)=q for x 0, limx→0+f(x)=(h+h2)1/2−h1/2h3/2=12∴f is continuous ⇒ p=−32, q=12

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The value of p and q for which the function $f (x) =$ $\{\begin{cases} \frac{\sin (p + 1) x + \sin x}{x}; & x < 0 \\ q; & x = 0 \\ \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}}; & x > 0 \end{cases}$ is continuous for all real values of x are

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

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

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

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

answer is D.

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Detailed Solution

$\begin{array}{l} a t x = 0, f (0) = q f o r x < 0, \lim_{x \to \bar{0}} f (x) \\ \lim_{h \to 0} \frac{\sin (P + 1) (0 - h) + \sin (0 - h)}{0 - h} = (P + 1) + 1 \\ = P + 2 \\ f o r x > 0, \lim_{x \to 0^{+}} f (x) = \frac{{(h + h^{2})}^{1 / 2} - h^{1 / 2}}{h^{3 / 2}} = \frac{1}{2} \\ ∴ f i s c o n t i n u o u s \Rightarrow p = - \frac{3}{2}, q = \frac{1}{2} \end{array}$