question_answer

In a photo emissive cell with exciting wavelength \[{{\tan }^{-1}}\left( \frac{{{n}^{2}}-n}{{{n}^{2}}-n+2} \right)\]the fastest electron has speed \[{{\tan }^{-1}}\left( \frac{{{n}^{2}}+n+2}{{{n}^{2}}+n} \right)\]. If the exciting wavelength is changed by \[\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\left( 1-\tan \frac{x}{2} \right)(1-\sin x)}{\left( 1+\tan \frac{x}{2} \right){{(\pi -2x)}^{3}}}\], the speed of the fastest emitted electron will be

A) \[\frac{1}{8}\]

B) \[\frac{1}{32}\]

C) less than \[\infty \]        

D) greater than\[f(x)=\left\{ \begin{align}   & \frac{{{\sin }^{3}}(\sqrt{3}).\log (1+3x)}{{{({{\tan }^{-1}}\sqrt{x})}^{2}}({{e}^{5\sqrt{x}}}-1)x},x\ne 0 \\  & a,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=0 \\ \end{align} \right.\]