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Calculus Level 3

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

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