Question

$\Rightarrow \lim_{x \rightarrow 0}\mspace{2mu}\frac{ln(sec(ex)) + ln\left( sec\left( e^{2}x \right) \right) + \ldots..\mathcal{l}n\left( sec\left( e^{10}x \right) \right)}{e^{2cosx}\left( \frac{e^{2 - 2cosx} - 1}{2 - 2cosx} \right) \times \frac{2 - 2cosx}{x^{2}} \times x^{2}}$

$$\Rightarrow \lim_{x \rightarrow 10}\mspace{2mu}\frac{\mathcal{l}n(sec(ex)) + \mathcal{l}n\left( sec\left( e^{2}x \right) \right) + \ldots\ldots\mathcal{l}n\left( sec\left( e^{10}x \right) \right)}{e^{2}x^{2}} $$Using L'H rule

$${\Rightarrow \lim_{x \rightarrow 10}\mspace{2mu}\frac{etanex + e^{2}\tan^{2}x + \ldots.. + e^{10}\tan^{210}x}{2e^{2}x} }{\Rightarrow \frac{1}{2e^{2}}\left\lbrack e^{2} + e^{4} + e^{6} + \ldots. + e^{20} \right\rbrack }{\Rightarrow \frac{1}{2}\frac{e^{2}\left( \left( e^{2} \right)^{10} - 1 \right)}{e^{2}\left( e^{2} - 1 \right)} }{\Rightarrow \frac{1}{2}\frac{\left( e^{20} - 1 \right)}{\left( e^{2} - 1 \right)} }$$