Question
यदि $f(x)=\left\{\begin{array}{ll}\frac{\sin (a+2) x+\sin x}{x} & ; x < 0 \\ b & ; x=0 \\ \frac{\left(x+3 x^{2}\right)^{\frac{1}{3}}-x^{-\frac{1}{3}}}{x^{\frac{4}{3}}} & ; x > 0\end{array}\right.$ $x =0$ पर संतत है, तो $a +2 b$ का मान है
$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0} \frac{\left(x+3 x^{2}\right)^{1 / 3}-x^{1 / 3}}{x^{4 / 3}}$
$=\lim _{x \rightarrow 0} \frac{(1+3 x)^{1 / 3}-1}{x}=1$
$f(0)=b$
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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