Question

Solve:
$\underset{\theta \to \pi }{lim}\frac{\pi -\theta }{\sqrt{\cos \theta +1}}$

Answer 1

We have,

${lim}_{\theta \to \pi }\frac{\pi -\theta }{\sqrt{\cos \theta +1}}$ $\left(\frac{0}{0}form\right)$

So,

$\Rightarrow {lim}_{\theta \to \pi }\frac{\pi -\theta }{\sqrt{2{\cos }^2\frac{\theta }{2}-1+1}}$

$\Rightarrow {lim}_{\theta \to \pi }\frac{\pi -\theta }{\sqrt{2}\cos \frac{\theta }{2}}$

Applying L’ Hospital rule and we get,

${lim}_{\theta \to \pi }\frac{0-1}{\sqrt{2}(-\sin \frac{\theta }{2})\times \frac{1}{2}}$

Taking limit and we get,

$\frac{1}{\frac{\sqrt{2}}{2}\sin \frac{\pi }{2}}$

$=\frac{2}{\sqrt{2}}or\sqrt{2}$

Hence, this is the answer.