If $f(x)=\left\{\begin{array}{cc}\frac{\sin (a+2) x+\sin x}{x} & ; x<0 \\ b & ; x=0 \\ \frac{\left(x+3 x^{2}\right)^{1 / 3}-x^{1 / 3}}{x^{4 / 3}} & ; x>0\end{array}\right.$
is continuous at $x=0$, then $a+2 b$ is equal to:
Correct Option: , 3
$\mathrm{LHL}=\lim _{x \rightarrow 0} \frac{\sin (a+2) x+\sin x}{x}$
$=\lim _{x \rightarrow 0}\left(\frac{\sin (a+2) x}{(a+2) x}\right)(a+2)+\lim _{x \rightarrow 0} \frac{\sin x}{x}=a+3$
$f(0)=b$
RHL $=\lim _{h \rightarrow 0}\left(\frac{(1+3 h)^{\frac{1}{3}}-1}{h}\right)=1$
$\because \quad$ Function $f(x)$ is continuous
$\therefore \quad \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)$
$\therefore \quad a+3=1 \Rightarrow a=-2$
and $b=1$
Hence, $a+2 b=0$
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