lim_(x→∞) cos((π/4))cos((π/8)) ... cos((π/2^(n+1) ))


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Question Number 174715 by infinityaction last updated on 09/Aug/22

$\underset{x \to \infty}{\lim c o s} (\frac{π}{4}) \cos (\frac{π}{8}) . . . \cos (\frac{π}{2^{n + 1}})$
	Answered by abdullahoudou last updated on 09/Aug/22

	$\cos (\frac{π}{2^{n}}) = \sin (\frac{π}{2^{n - 1}}) \frac{1}{2 \sin (π / 2^{n})}$ $t e l e s c o p e$
	Commented by infinityaction last updated on 09/Aug/22

	$c a n y o u t e l l m e f u l l s o l u t i o n$ $o f t h i s q u e s t i o n$
	Answered by princeDera last updated on 10/Aug/22
	$sin (2x) = 2cos (x)sin (x) cos (x) = ((sin(2x))/(sin (x))) cos ((π/2^n )) = ((sin (((2π)/2^n )))/(sin ((π/2^n )))) = ((sin ((π/2^(n−1) )))/(sin ((π/2^n )))) Ω = lim_(n→∞) Π_(k=2) ^(n+1) ((sin ((π/2^(n−1) )))/(sin ((π/2^n )))) = lim_(n→∞) { ((sin (π/2))/(sin (π/4)))×((sin((π/4)) )/(sin ((π/8))))× ×((sin ((π/2^(n−1) )))/(sin ((π/2^n ))))×((sin ((π/2^n )))/(sin ((π/2^(n+1) ))))} Ω = lim_(n→∞) {((sin ((π/2)))/(sin ((π/2^(n+1) ))))} As n→∞,sin {(π/2^(n+1) )} → 0 the limit doesn′t exist$
	$\sin (2 x) = 2 \cos (x) \sin (x)$ $\cos (x) = \frac{s i n (2 x)}{\sin (x)}$ $\cos (\frac{π}{2^{n}}) = \frac{\sin (\frac{2 π}{2^{n}})}{\sin (\frac{π}{2^{n}})} = \frac{\sin (\frac{π}{2^{n - 1}})}{\sin (\frac{π}{2^{n}})}$ $Ω = \lim_{n \to \infty} \underset{k = 2}{\prod^{n + 1}} \frac{\sin (\frac{π}{2^{n - 1}})}{\sin (\frac{π}{2^{n}})} = \lim_{n \to \infty} {\frac{\sin (π / 2)}{\sin \frac{π}{4}} \times \frac{\sin (\frac{π}{4})}{\sin (\frac{π}{8})} \times \times \frac{\sin (\frac{π}{2^{n - 1}})}{\sin (\frac{π}{2^{n}})} \times \frac{\sin (\frac{π}{2^{n}})}{\sin (\frac{π}{2^{n + 1}})}}$ $Ω = \lim_{n \to \infty} {\frac{\sin (\frac{π}{2})}{\sin (\frac{π}{2^{n + 1}})}}$ $A s n \to \infty, \sin {\frac{π}{2^{n + 1}}} \to 0$ $t h e l i m i t {d o e s n}^{'} t e x i s t$
	Commented by infinityaction last updated on 09/Aug/22

	$d o n t u s e l h o p i t a l r u l e$ $b e c a u s e n o t \frac{0}{0} o r \frac{\infty}{\infty} f o r m$
	Commented by princeDera last updated on 10/Aug/22

	$y e a h$ $t h e l i m i t {d o e s n}^{'} t e x i s t$
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