If f(x) = \ cases ( (a + 2)x + x)/(x) & ; x 0 cases ) is continuous...

f\left({0}^{-}\right) = \lim_{x \to {0}^{-}}\left(\dfrac{sinaxcos2x}{x} + \dfrac{\sin 2xcosax}{x} + \dfrac{\sin x}{x}\right)$</p><p>$= \lim_{x \to {0}^{-}}\left(acos2x + 2cosax + 1\right)$</p><p>$= a + 3$</p><p>Also,$f\left(0\right) = b$<br />$f\left({0}^{+}\right) = \lim_{x \to 0}\dfrac{{x}^{\dfrac{1}{3}}}{{x}^{\dfrac{1}{3}}}\left(\dfrac{\left(1 + 3x\right)\dfrac{1}{3} - 1}{x}\right)$, this is \left(\dfrac{0}{0}\right)$form

Apply $LH$ Rule

$⇒ f\left({0}^{+}\right) = 1$

$∴ a = - 2$

$b = 1$

$∴ a + 2b = 0$