Question

Solve :${lim}_{x\to 0}\left[\frac{{\log }_{\sec \left(\frac{x}{2}\right)}\cos x}{{\log }_{\sec x}\cos \left(x/2\right)}\right]$=

Answer 1

$\underset{x+0}{lim}\left[\frac{{\log }_{\sec \left(\frac{x}{2}\right)}\cos x}{{\log }_{\sec x}\cos \left(\frac{x}{2}\right)}\right]$

$\Rightarrow {\log }_{\sec \frac{x}{2}}\cos x=\frac{\log \cos x}{\log \sec \frac{x}{2}}$....... $\left(1\right)$ $({\log }_{b}a=\frac{\log a}{\log b})$

$\Rightarrow {\log }_{\sec x}\cos \frac{x}{2}=\frac{\log \left(\cos \frac{x}{2}\right)}{\log \left(\sec x\right)}$ ........... $\left(2\right)$

$\begin{array}{c}\left(1\right)\\ \left(2\right)\end{array}$ $=\frac{\log \left(\cos x\right)\times \log \left(\sec x\right)}{\log \sec \left(\frac{x}{2}\right)\times \left(\log \sec \frac{x}{2}\right)}$

$\Rightarrow {\left[\frac{(-1)\log \cos x}{(-1)\log \cos \left(\frac{x}{2}\right)}\right]}^2$

as $x\to 0$, it takes $\frac{0}{0}$ in determinate form them, applying L'hospital Rule

$\underset{x\Rightarrow 0}{lim}{[\left(\frac{-\sin x}{\cos x}\right)÷\left(\frac{\sin \frac{x}{2}}{2\cos \left(\frac{x}{2}\right)}\right)]}^2$

$\underset{x\Rightarrow 0}{lim}{\left(\frac{2\tan x}{\tan \frac{x}{2}}\right)}^2$

$\underset{x\Rightarrow 0}{lim}\left(\frac{4\left(\tan \frac{x}{2}\right)}{\tan \left(\frac{x}{2}\right)\times (1+{\tan }^2\frac{x}{2})}\right)$

$lim{\left(\frac{4}{1}\right)}^2$

$\Rightarrow 16$