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STD 12 - 5. continuity and differentiation — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 5. continuity and differentiation4 Marks
Question
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
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Answer

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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