Let \( f(x)=\max \{p, q, r\} \), where
\[
p=\lim _{n \rightarrow \infty} \lim _{\alpha \rightarr...
Let \( f(x)=\max \{p, q, r\} \), where
\[
p=\lim _{n \rightarrow \infty} \lim _{\alpha \rightarrow 1^{+}} \frac{\alpha^{n}|\sin x|+\alpha^{-n}|\cos x|}{\alpha^{n}+\alpha^{-n}}
\]
\( q=\lim _{x \rightarrow \infty} \lim _{\alpha \rightarrow 1^{-}} \frac{\alpha^{n}|\sin x|+\alpha^{-n}|\cos x|}{\alpha^{n}+\alpha^{-n}} \)
\( r=\lim _{n \rightarrow \infty} \frac{\pi}{4 n}\left[1+\cos \frac{\pi}{2 n}+\cos \frac{2 \pi}{2 n}+\ldots+\cos \frac{(n-1) \pi}{2 n}\right] \). Then
The value of \( p \) is
(a) \( 2|\sin x| \)
(b) \( |\cos x| \)
(c) \( |\sin x| \)
(d) \( \frac{1}{2} \)
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