Question

Evaluate following limits
${lim}_{X\to \frac{\pi }{2}}\frac{2^{-\cos x}-1}{x(x-\frac{\pi }{2})}$

Answer 1

$\underset{x\to \frac{\pi }{2}}{lim}\frac{2^{-\cos x}-1}{x\left(x-\frac{\pi }{2}\right)}\left(\frac{0}{0}form\right)$

$=\underset{x\to \frac{\pi }{2}}{lim}\frac{2^{-\cos x}\log 2\times \sin x}{2x-\frac{\pi }{2}}$

$=\frac{\log 2}{\pi -\frac{\pi }{2}}$

$=\frac{2}{\pi }\log 2$