If f(x)=

Question:

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:

 

  1. (1) 1

  2. (2) $-1$

  3. (3) 0

  4. (4) $-2$


Correct Option: , 3

Solution:

$\mathrm{LHL}=\lim _{x \rightarrow 0} \frac{\sin (a+2) x+\sin x}{x}$

$=\lim _{x \rightarrow 0}\left(\frac{\sin (a+2) x}{(a+2) x}\right)(a+2)+\lim _{x \rightarrow 0} \frac{\sin x}{x}=a+3$

$f(0)=b$

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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