If f(x)= ll (a+2) x+ x x & ; x<0 b & ; x=0 (x+3 x^ 2 )^ 1 3 -x^ 1… - Vidyadip

MCQ

If $f(x)=\left\{\begin{array}{ll}{\frac{\sin (a+2) x+\sin x}{x}} & {; x<0} \\ {b} & {; x=0} \\ {\frac{\left(x+3 x^{2}\right)^{\frac{1}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}}} & {; x>0}\end{array}\right.$ is continuous at $x=0,$ then $a+2 b$ is equal to

A
$-1$
B
$1$
C
$-2$
✓
$0$

✓

Answer

Correct option: D.

$0$

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

$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0} \frac{\left(x+3 x^{2}\right)^{1 / 3}-x^{1 / 3}}{x^{4 / 3}}$

$=\lim _{x \rightarrow 0} \frac{(1+3 x)^{1 / 3}-1}{x}=1$

$f(0)=b$

for continuity at $x=0$ $\lim _{x \rightarrow 0^{-}} f(x)=f(0)=\lim _{x \rightarrow 0^{+}} f(x)$

$\Rightarrow \quad a+3=b=1$

$\therefore \quad a=-2, \quad b=1$

$\therefore \quad a+2 b=0$

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