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

is continuous at $x = 0,$ then $k =$

Solution:

N Since  $f \left(x\right)$  is continuous at  $x = 0.$ Therefore,

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

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