The value of p for which the function $f(x)=\left\{\begin{matrix}\frac{(4^x-1)^3}{\sin\frac{x}{p}\log e[1+\frac{x^2}{3}]};&x≠0\\12(\log_e4)^3;&x=0\end{matrix}\right.$ may be continuous at x = 0, is

none of these

For f(x) to be continuous at x = 0, we have

$\underset{x→0}{\lim}f(x)=f(0)=12(\log 4)^3$

$\underset{x→0}{\lim}f(x)=\underset{x→0}{\lim}(\frac{4^x-1}{x})^3×\frac{(\frac{x}{p})}{(\sin\frac{x}{p})}.\frac{px^2}{\log(1+\frac{1}{3}x^2)}$

$=(\log 4)^3.1.p.\underset{x→0}{\lim}(\frac{x^2}{\frac{1}{3}x^2-\frac{1}{18}x^4+....})=3p(\log 4)^3$

Hence, p = 4