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

$\left(\log _e 4\right)^3$

Since f(x) is continuous at x = 0. Therefore,

$\lim\limits_{x \rightarrow 0} f(x)=f(0)$

$\Rightarrow \lim\limits_{x \rightarrow 0} \frac{\left(4^x-1\right)^3}{\sin \frac{x}{4} \log \left(1+\frac{x^2}{3}\right)}=k$

$\Rightarrow \lim\limits_{x \rightarrow 0} \frac{12\frac{\left(4^x-1\right)^3}{x}}{\left(\frac{\sin \frac{x}{4}}{x / 4}\right)\left(\frac{\log \left(1+x^2 / 3\right)}{x^2 / 3}\right)}=k$

$\Rightarrow 12(\log 4)^3=k \Rightarrow k=96(\log 2)^3$