If $$f(x)=\left\{\begin{matrix} \dfrac{\sin (a+2)x+\sin x}{x}; & x < 0\\ b; & x=0\\ \dfrac{(x+3x^2)^{1/3}-x^{1/3}}{x^{4/3}}; & x > 0\end{matrix}\right.$$
is continuous at $$x=0$$, then $$a+2b$$ is equal to?
Correct option is D. $$0$$
For LHL
$$LHL = \underset{x \rightarrow 0}{\lim} \dfrac{\sin (a + 2) x + \sin x}{x}$$
$$= \underset{x \rightarrow 0}{\lim} \dfrac{2 \sin \left(\dfrac{a + 3}{2}\right) x \cos \left(\dfrac{(a + 1)x}{2} \right)}{x}$$
$$= \underset{x \rightarrow 0}{\lim} \dfrac{2 \sin \left(\dfrac{a + 3}{2} \right) x \cos \left(\dfrac{a + 1}{2} \right) x}{x}$$
$$LHL = a +3$$
$$RHL = \underset{x \rightarrow 0}{\lim} \dfrac{(x + 3x^2)^{1/3} - x^{1/3}}{x^{4/3}}$$
$$= \underset{h \rightarrow 0}{\lim} \dfrac{((1 + 3h)^{1/3} - 1)}{h}$$
$$= 1$$
$$f(0) = b$$
So, $$a = -2 , b = 1$$
$$\Rightarrow a + 2b = 0$$
$$\therefore$$ $$\boxed{ a + 2b = 0}.....Answer$$
Hence option $$'D'$$ is the answer.