The value of \(\lim _{x\to 0}\frac{{\log }_{e}\left(\sec \left(ex\right)⋅\sec \left({e}^{2}x\right)⋅\ldots ⋅\sec \left({…

\(L=\lim _{x\to 0}\frac{{\log }_{e}\left(\sec \left(ex\right)⋅\sec \left({e}^{2}x\right)⋅\ldots ⋅\sec \left({e}^{10}x\right)\right)}{{e}^{2}−{e}^{2\cos x}}\)

\(=\lim _{x\to 0}\frac{\ln \left(\sec \left(ex\right)\right)+\ln \left(\sec \left({e}^{2}x\right)\right)+\ldots ..+\ln \left(\sec \left({e}^{10}x\right)\right)}{{e}^{2\cos x}\left(\frac{{e}^{2−2\cos x}−1}{2−2\cos x}\right)\times \frac{2−2\cos x}{{x}^{2}}\times {x}^{2}}\)

\(=\lim _{x\to 0}\frac{\ln \left(\sec \left(ex\right)\right)+\ln \left(\sec \left({e}^{2}x\right)\right)+\ldots \ldots +\ln \left(\sec \left({e}^{10}x\right)\right)}{{e}^{2}{x}^{2}}\)

Using L'H rule

\(=\lim _{x\to 0}\frac{e\tan \left(ex\right)+{e}^{2}\tan \left({e}^{2}x\right)+\ldots ..+{e}^{10}\tan \left({e}^{10}x\right)}{2{e}^{2}x}\)

\(=\frac{1}{2{e}^{2}}\left[{e}^{2}+{e}^{4}+{e}^{6}+\ldots .+{e}^{20}\right]\)

\(=\frac{1}{2}\frac{{e}^{2}\left({\left({e}^{2}\right)}^{10}−1\right)}{{e}^{2}\left({e}^{2}−1\right)}\)

\(=\frac{1}{2}\frac{\left({e}^{20}−1\right)}{\left({e}^{2}−1\right)}\)