If f(x)={sin(p+1)x+sinxx,q,x+x2−xx3/2,x>0x=0x>0 is continuous at x = 0, then the ordered (p,q)pair is equal to :

RHL=limx→0+x+x2−xx3/2=limx→0+1+x−1x=12,LHL=limx→0−sin(p+1)x+sinxx=(p+1)+1=p+2For function to be continuous LHL=RHL=f(0)⇒(p,q)=(−32,12)

If f(x)={sin(p+1)x+sinxx,q,x+x2−xx3/2,x>0x=0x>0 is continuous at x = 0, then the ordered (p,q)pair is equal to :

RHL=limx→0+x+x2−xx3/2=limx→0+1+x−1x=12,LHL=limx→0−sin(p+1)x+sinxx=(p+1)+1=p+2For function to be continuous LHL=RHL=f(0)⇒(p,q)=(−32,12)

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If $f (x) = {\begin{matrix} \frac{\sin (p + 1) x + \sin x}{x}, \\ q, \\ \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}}, \end{matrix} \begin{matrix} x > 0 \\ x = 0 \\ x > 0 \end{matrix}$ is continuous at x = 0, then the ordered $(p, q)$ pair is equal to :

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

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

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

$(\frac{5}{2}, \frac{1}{2})$

answer is C.

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Detailed Solution

$R H L = \lim_{x \to 0^{+}} \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}}$

$= \lim_{x \to 0^{+}} \frac{\sqrt{1 + x} - 1}{x} = \frac{1}{2}, L H L = \lim_{x \to 0^{-}} \frac{\sin (p + 1) x + \sin x}{x} = (p + 1) + 1 = p + 2$ For function to be continuous $L H L = R H L = f (0) \Rightarrow (p, q) = (- \frac{3}{2}, \frac{1}{2})$