Question

The value of $\underset{x\to \pi /4}{lim}\frac{\left(\sqrt{2}-\cos \theta -\sin \theta \right)}{{\left(4\theta -\pi \right)}^2}$ is

• A. $\frac{1}{16\sqrt{2}}$
• B. $\frac{1}{16}$
• C. $\frac{1}{8\sqrt{2}}$
• D. $\frac{1}{2\sqrt{2}}$

Answer 1

The correct option is A $\frac{1}{16\sqrt{2}}$

$\underset{x\to \frac{\pi }{4}}{lim}\frac{\sqrt{2}-\cos \theta -\sin \theta }{{(4\theta -\pi )}^2}=\frac{0}{0}$ form

Applying L-Hospital's rule

$\underset{x\to \frac{\pi }{4}}{lim}\frac{\sin \theta -\cos \theta }{2(4\theta -\pi )\left(4\right)}=\frac{0}{0}$ form

$=\underset{x\to \frac{\pi }{4}}{lim}\frac{\cos \theta +\sin \theta }{8\left(4\right)}$

$=\frac{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}{32}=\frac{2}{\sqrt{2}}\times \frac{1}{32}=\frac{\sqrt{2}}{32}=\frac{1}{16\sqrt{2}}$