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Calculus Level 3

limx0(sin(x)tan(5x)×(1cos(4x))(5x4x)x3)=qrln(rt)\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 q\mathfrak{q}, r\mathfrak{r} and t\mathfrak{t} with r\mathfrak{r} and t\mathfrak{t} being coprime integers.

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

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