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$\lim_{x\rightarrow 0} \left(\dfrac{\sin(x)}{\tan(5x)}\times \dfrac{(1-\cos(4x))(5^x-4^x)}{x^3}\right) =\dfrac{\mathfrak{q}}{\mathfrak{r}}\ln\left(\dfrac{\mathfrak{r}}{\mathfrak{t}}\right)$

The above equation is true for positive integers $\mathfrak{q}$, $\mathfrak{r}$ and $\mathfrak{t}$ with $\mathfrak{r}$ and $\mathfrak{t}$ being coprime integers.

Find $\mathfrak{q}+\mathfrak{r}+\mathfrak{t}$.