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Q.

The value of a and b for which the function $F (x) =$ $\{\begin{cases} \frac{\sin (a + 1) x + \sin x}{x} & x < 0 \\ b & x = 0 \\ \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}} & \begin{matrix} x > 0 \end{matrix} \end{cases}$ is continous for all x.

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a

$a = \frac{- 3}{2}, b = \frac{1}{2}$

b

$a = \frac{5}{2}, b = \frac{1}{2}$

c

$a = \frac{1}{2}, b = \frac{3}{2}$

d

$a = \frac{1}{2}, b = - \frac{3}{2}$

answer is A.

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Detailed Solution

$\lim_{x \to 0} \frac{\sin (a + 1) x + \sin x}{x} = a + 2$

$\lim_{x \to 0} \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}} = \lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x} = \frac{1}{2}$

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