The value of p and q for which the function is continuous for all x in R, are:

Engineering

Mathematics

Basic Definition of Continuity

Question

The value of p and q for which the function $f (x) = \{\begin{array}{cl} \frac{\sin (p + 1) x + \sin x}{x}, & x < 0 \\ q, & x = 0 \\ \frac{\sqrt{x + x^{2}} - \sqrt{x}}{x^{3 / 2}}, & x > 0 \end{array}$ is continuous for all x in R, are:

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

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

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

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

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f(0) = q

$f (0^{+}) = lim_{x \to 0^{+}} \frac{(1 + x)^{1 / 2} - 1}{x} = lim_{x \to 0^{+}} \frac{1 + \frac{1}{2} x + \dots \dots . - 1}{x} = \frac{1}{2}$

$f (0^{-}) = lim_{x \to 0^{-}} \frac{\sin (p + 1) x + \sin x}{x}$

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

= (p + 1) + 1 = p + 2

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