Question

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

Answer 1

${lim}_{\theta \to \frac{\pi }{4}}\frac{\sqrt{2}-cos\theta -sin\theta }{(4\theta -\pi {)}^2}$

On Substituting $\theta =\frac{\pi }{4}$, we can see that the expression is in $\frac{0}{0}$ form

So here we apply $L^{\prime }Hospita{l}^{\prime }sRule$

$\Rightarrow {lim}_{\theta \to \frac{\pi }{4}}\frac{\frac{d}{d\theta }(\sqrt{2}-cos\theta -sin\theta )}{\frac{d}{d\theta }(4\theta -\pi {)}^2}$

$\Rightarrow {lim}_{\theta \to \frac{\pi }{4}}\frac{0-(-sin\theta )-cos\theta }{2(4\theta -\pi )\times 4}$

$\Rightarrow {lim}_{\theta \to \frac{\pi }{4}}\frac{sin\theta -cos\theta }{8(4\theta -\pi )}$

On Substituting $\theta =\frac{\pi }{4}$, we can see that the expression is still in $\frac{0}{0}$ form

So here we again apply $L^{\prime }Hospita{l}^{\prime }sRule$

$\Rightarrow {lim}_{\theta \to \frac{\pi }{4}}\frac{\frac{d}{d\theta }(sin\theta -cos\theta )}{\frac{d}{d\theta }\left(8\right(4\theta -\pi \left)\right)}$

$\Rightarrow {lim}_{\theta \to \frac{\pi }{4}}\frac{cos\theta -(-sin\theta )}{(8\times 4)}$

$\Rightarrow {lim}_{\theta \to \frac{\pi }{4}}\frac{cos\theta +sin\theta }{32}$

Now substitute $\theta =\frac{\pi }{4}$

$\Rightarrow \frac{cos\frac{\pi }{4}+sin\frac{\pi }{4}}{32}$

$\Rightarrow \frac{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}{32}$

$\Rightarrow \frac{\frac{2}{\sqrt{2}}}{32}=\frac{1}{16\sqrt{2}}$

Therefore, $\Rightarrow {lim}_{\theta \to \frac{\pi }{4}}\frac{sin\theta -cos\theta }{8(4\theta -\pi )}=\frac{1}{16\sqrt{2}}$