If f(x)=4x−13sin⁡(x/4)log⁡1+x2/3,x≠0k,x=0 is continuous at  x = 0, then  k =

# If $f\left(x\right)=\left\{\begin{array}{cc}\frac{{\left({4}^{x}-1\right)}^{3}}{\mathrm{sin}\left(x/4\right)\mathrm{log}\left(1+{x}^{2}/3\right)},& x\ne 0\\ k& ,x=0\end{array}\right\$ is continuous at  then

1. A

$12{\left({\mathrm{log}}_{e}4\right)}^{2}$

2. B

$96\left(\mathrm{log}2{\right)}^{3}$

3. C

${\left({\mathrm{log}}_{e}4\right)}^{3}$

4. D

none of these

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### Solution:

N Since    is continuous at   Therefore,

$\begin{array}{r}\underset{x\to 0}{lim} f\left(x\right)=f\left(0\right)\\ ⇒\underset{x\to 0}{lim} \frac{{\left({4}^{x}-1\right)}^{3}}{\mathrm{sin}\frac{x}{4}\mathrm{log}\left(1+\frac{{x}^{2}}{3}\right)}=k\\ ⇒\underset{x\to 0}{lim} \frac{12\left(\frac{{4}^{x}-{1}^{3}}{x}\right)}{\left(\frac{\mathrm{sin}\frac{x}{4}}{x/4}\right)\left(\frac{\mathrm{log}\left(1+{x}^{2}/3\right)}{{x}^{2}/3}\right)}=k\end{array}$

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