Let f : R $\to$ R be a continuously differentiable function such that f(2) = 6 and  $f^{\prime }\left(2\right)=\frac{1}{48}$ . If  ${\int }_6^{f\left(x\right)}4t^{3}dt=\left(x-2\right)g\left(x\right),$, then $\underset{x\to 2}{\lim }g\left(x\right)$ g(x) is equal to :