Question

Find the value of $f\left(0\right)$ so that the function $f\left(x\right)=\frac{{(4^x-1)}^3}{\sin \frac{x}{4}\log (1+x^2/3){\left(\log 2\right)}^3},x\ne 0$ is continuous

Answer 1

$f\left(0\right)=\underset{x\to 0}{lim}\frac{{(4^x-1)}^3}{\sin \frac{x}{4}\log (1+x^3/3){\left(\log 2\right)}^3}$

$=12.\underset{x\to 0}{lim}\frac{{\left(\frac{4^x-1}{x}\right)}^3}{\frac{\sin \frac{x}{4}}{x/4}.\frac{\log (1+x^2/3)}{x^2/3}{\left(\log 2\right)}^3}$

$=12.\frac{(\log 4{)}^3}{(\log 2{)}^3}=12\cdot 8=96$