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limx→0(sin(x)tan(5x)×(1−cos(4x))(5x−4x)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)x→0lim(tan(5x)sin(x)×x3(1−cos(4x))(5x−4x))=rqln(tr)
The above equation is true for positive integers q\mathfrak{q}q, r\mathfrak{r}r and t\mathfrak{t}t with r\mathfrak{r}r and t\mathfrak{t}t being coprime integers.
Find q+r+t\mathfrak{q}+\mathfrak{r}+\mathfrak{t}q+r+t.
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