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
Range of \( f(x) \) is
(a) \( [0,1] \)
(b) \( \left[\frac{1}{2}, 1\right] \)
(c) \( \left[\frac{1}{\sqrt{2}}, 1\right] \)
(d) \( \left[\frac{1}{2}, 2\right] \)
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