Question

Solve $y=\underset{x\to \frac{\pi }{2}-}{lim}{\left(\sec x\right)}^{\cot x}$

Answer 1

${lim}_{x\to \frac{\pi }{2}-}\left({\left(\sec \left(x\right)\right)}^{\cot \left(x\right)}\right)$

$Applyexponentrule:a^x=e^{\ln \left(a^x\right)}=e^{x\cdot \ln \left(a\right)}$

$={lim}_{x\to \frac{\pi }{2}-}\left(e^{\cot \left(x\right)\ln \left(\sec \left(x\right)\right)}\right)$

$={lim}_{x\to \frac{\pi }{2}-}\left(e^{\cot \left(x\right)\ln \left(\sec \left(x\right)\right)}\right)$

${lim}_{x\to \frac{\pi }{2}-}\left(\cot \left(x\right)\ln \left(\sec \left(x\right)\right)\right)$

$={lim}_{x\to \frac{\pi }{2}-}\left(\frac{\ln \left(\sec \left(x\right)\right)}{\frac{1}{\cot \left(x\right)}}\right)$

Apply L-Hospital's rule

$={lim}_{x\to \frac{\pi }{2}-}\left(\frac{\tan \left(x\right)}{{\sec }^2\left(x\right)}\right)$

$={lim}_{x\to \frac{\pi }{2}-}\left(\sin \left(x\right)\cos \left(x\right)\right)$

$=\sin \left(\frac{\pi }{2}\right)\cos \left(\frac{\pi }{2}\right)$

$=0$

${lim}_{u\to 0-}\left(e^u\right)=1$

${lim}_{x\to \frac{\pi }{2}-}\left({\left(\sec \left(x\right)\right)}^{\cot \left(x\right)}\right)=1$