First slide

Continuity

Question

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

Difficult

A

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

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

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

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

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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