Determine `f(0)` so that the function `f(x)` defined by `f(x)=((4^x-1)^3)/(sinx/4log(1+(x^2)/3))` becomes continuous at `x=0`

Explanation

`f(x) =lim _{x rightarrow 0} frac{(4^{x}-1)^{3} 1}{sin (frac{x}{4}) log (1+frac{x^{2}}{3})} `

`
=lim _{x rightarrow 0}(frac{4^{x}-1}{x})^{3} frac{4 cdot frac{x}{4}}{sin (frac{x}{4})} cdot frac{1}{frac{1}{x^{2}} log (1+frac{x^{2}}{3})} `

`
=lim _{x rightarrow 0}(frac{4^{x}-1}{x})^{3} cdot 4 cdot(frac{frac{x}{4}}{sin frac{x}{4}}) cdot frac{1}{frac{1}{3} log (1+frac{x^{2}}{3})^{3 / x^{2}}} `

`=(log 4)^{3} cdot 4 cdot 3 cdot frac{1}{1}=12(log 4)^{3} `