$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{1+\cos 2x}{cot3x(3^{\sin 2x}-1)}=$

$$\begin{gathered} \underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\mathrm{Lt}}\frac{1+\cos 2\mathrm{x}}{\cot 3\mathrm{x}(3^{\sin 2\mathrm{x}}-1)} \\ =\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\mathrm{Lt}}\frac{2{\cos }^2\mathrm{x} \tan 3\mathrm{x}}{3^{\sin 2\mathrm{x}}-1} \\ =\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\mathrm{Lt}}\frac{2 {\sin }^2\left({\displaystyle \frac{\mathrm{\pi }}{2}}-x\right)\tan x\left({\displaystyle \frac{3-{\tan }^{2}x}{1-3{\tan }^{2}x}}\right)}{3^{\sin {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\mathrm{\pi }}{2}}-x\right)}-1} \\ =\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\mathrm{Lt}}\frac{2 {\sin }^2\left({\displaystyle \frac{\mathrm{\pi }}{2}-x}\right){\displaystyle \frac{\cos \left({\displaystyle \frac{\mathrm{\pi }}{2}}-x\right)}{\sin \left(\frac{\mathrm{\pi }}{2}-x\right)}}\left({\displaystyle \frac{3-{\tan }^{2}x}{1-3{\tan }^{2}x}}\right)}{3^{\sin 2\left({\displaystyle \frac{\mathrm{\pi }}{2}-x}\right)}-1} \\ =\frac{3}{\log 3} \end{gathered}$$