Step 1: Simplify the expression
The given limit is \(\lim _{x\rightarrow \pi }\frac{1-\sin \frac{x}{2}}{\cos \frac{x}{2}\left(\cos \frac{x}{4}-\sin \frac{x}{4}\right)}\).
First, we can simplify the numerator using the identity \(\sin (2\theta )=2\sin \theta \cos \theta \) and \(\sin ^{2}\theta +\cos ^{2}\theta =1\).
We can rewrite \(1\) as \(\sin ^{2}\frac{x}{4}+\cos ^{2}\frac{x}{4}\) and \(\sin \frac{x}{2}\) as \(2\sin \frac{x}{4}\cos \frac{x}{4}\).
So, the numerator becomes:\(1-\sin \frac{x}{2}=\sin ^{2}\frac{x}{4}+\cos ^{2}\frac{x}{4}-2\sin \frac{x}{4}\cos \frac{x}{4}=\left(\cos \frac{x}{4}-\sin \frac{x}{4}\right)^{2}\)
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Step 2: Substitute the simplified numerator into the limit Substitute the simplified numerator back into the original expression:\(\lim _{x\rightarrow \pi }\frac{\left(\cos \frac{x}{4}-\sin \frac{x}{4}\right)^{2}}{\cos \frac{x}{2}\left(\cos \frac{x}{4}-\sin \frac{x}{4}\right)}\)We can cancel out one factor of \(\left(\cos \frac{x}{4}-\sin \frac{x}{4}\right)\) from the numerator and denominator:\(\lim _{x\rightarrow \pi }\frac{\cos \frac{x}{4}-\sin \frac{x}{4}}{\cos \frac{x}{2}}\)
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Step 3: Evaluate the limit Now, we can substitute \(x=\pi \) into the simplified expression:\(\frac{\cos \frac{\pi }{4}-\sin \frac{\pi }{4}}{\cos \frac{\pi }{2}}=\frac{\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}}{0}=\frac{0}{0}\)The expression is still an indeterminate form. We need to simplify it further.
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Step 4: Further simplification using trigonometric identities We can simplify the numerator \(\cos \frac{x}{4}-\sin \frac{x}{4}\).
Multiply and divide by \(\sqrt{2}\):\(\sqrt{2}\left(\frac{1}{\sqrt{2}}\cos \frac{x}{4}-\frac{1}{\sqrt{2}}\sin \frac{x}{4}\right)\)
Using the identities \(\sin \frac{\pi }{4}=\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}}\) and \(\cos (A+B)=\cos A\cos B-\sin A\sin B\),
the numerator becomes:\(\sqrt{2}\left(\cos \frac{\pi }{4}\cos \frac{x}{4}-\sin \frac{\pi }{4}\sin \frac{x}{4}\right)=\sqrt{2}\cos \left(\frac{\pi }{4}+\frac{x}{4}\right)\)The denominator is \(\cos \frac{x}{2}\). Using the identity \(\cos A=\sin \left(\frac{\pi }{2}-A\right)\),
we can write:\(\cos \frac{x}{2}=\sin \left(\frac{\pi }{2}-\frac{x}{2}\right)=\sin \left(\frac{\pi -x}{2}\right)\)The expression now is:\(\lim _{x\rightarrow \pi }\frac{\sqrt{2}\cos \left(\frac{\pi }{4}+\frac{x}{4}\right)}{\sin \left(\frac{\pi -x}{2}\right)}=\lim _{x\rightarrow \pi }\frac{\sqrt{2}\cos \left(\frac{\pi +x}{4}\right)}{\sin \left(\frac{\pi -x}{2}\right)}\)
Step 5: Final evaluation of the limit Let \(y=x-\pi \). As \(x\rightarrow \pi \), \(y\rightarrow 0\).
So, \(x=y+\pi \).Substitute \(x=y+\pi \) into the expression:\(\lim _{y\rightarrow 0}\frac{\sqrt{2}\cos \left(\frac{\pi +(y+\pi )}{4}\right)}{\sin \left(\frac{\pi -(y+\pi )}{2}\right)}=\lim _{y\rightarrow 0}\frac{\sqrt{2}\cos \left(\frac{2\pi +y}{4}\right)}{\sin \left(\frac{-y}{2}\right)}\)Using \(\cos (A+B)=\cos A\cos B-\sin A\sin B\) and \(\sin (-A)=-\sin A\):\(\lim _{y\rightarrow 0}\frac{\sqrt{2}\left(\cos \left(\frac{\pi }{2}\right)\cos \left(\frac{y}{4}\right)-\sin \left(\frac{\pi }{2}\right)\sin \left(\frac{y}{4}\right)\right)}{-\sin \left(\frac{y}{2}\right)}\)\(=\lim _{y\rightarrow 0}\frac{\sqrt{2}\left(0\cdot \cos \left(\frac{y}{4}\right)-1\cdot \sin \left(\frac{y}{4}\right)\right)}{-\sin \left(\frac{y}{2}\right)}=\lim _{y\rightarrow 0}\frac{-\sqrt{2}\sin \left(\frac{y}{4}\right)}{-\sin \left(\frac{y}{2}\right)}=\lim _{y\rightarrow 0}\frac{\sqrt{2}\sin \left(\frac{y}{4}\right)}{\sin \left(\frac{y}{2}\right)}\)
Using the small angle approximation \(\sin \theta \approx \theta \) as \(\theta \rightarrow 0\):\(\lim _{y\rightarrow 0}\frac{\sqrt{2}\left(\frac{y}{4}\right)}{\left(\frac{y}{2}\right)}=\frac{\sqrt{2}}{4}\cdot 2=\frac{\sqrt{2}}{2}\)
Answer: The limit is \(\frac{\sqrt{2}}{2}\).