Question

Solve $\underset{x\to \frac{\pi }{2}}{lim}\frac{2^{-\cos x-1}}{x(x-\frac{\pi }{2})}$

Answer 1

$\underset{x\to \frac{\pi }{2}}{lim}\frac{2^{-cosx}-1}{x(x-\frac{\pi }{2})}$

$\underset{x\to \frac{\pi }{2}}{lim}\frac{2^{-cosx}-1}{x^2-\frac{\pi x}{2}}$

This is in $\frac{0}{0}$ form. So, using L'Hospital's rule we get

$\Rightarrow \underset{x\to \frac{\pi }{2}}{lim}\frac{2^{-cosx}\left(ln2\right)sinx}{2x-\frac{\pi }{2}}$

Substituting $x=\frac{\pi }{2}$ we get

$\Rightarrow \frac{2ln2}{\pi }$