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lim-n-2-n-sin-pi-2-n-




Question Number 103749 by bobhans last updated on 17/Jul/20
lim_(n→∞)  2^n  sin ((π/2^n )) = ?
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{2}^{{n}} \:\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}^{{n}} }\right)\:=\:? \\ $$
Answered by bramlex last updated on 17/Jul/20
set x = (π/2^n ) ⇒2^n  = (π/x) ;x→0  lim_(x→0)  (π/x).sin x = π lim_(x→0)  ((sin x)/x) = π .  □
$${set}\:{x}\:=\:\frac{\pi}{\mathrm{2}^{{n}} }\:\Rightarrow\mathrm{2}^{{n}} \:=\:\frac{\pi}{{x}}\:;{x}\rightarrow\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\pi}{{x}}.\mathrm{sin}\:{x}\:=\:\pi\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:{x}}{{x}}\:=\:\pi\:. \\ $$$$\square \\ $$
Answered by Sobir last updated on 17/Jul/20
the  ansver is   π
$${the}\:\:{ansver}\:{is}\:\:\:\pi \\ $$
Answered by Dwaipayan Shikari last updated on 17/Jul/20
lim_(n→∞) π((sin(π/2^n ))/(π/2^n ))=π
$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\pi\frac{{sin}\frac{\pi}{\mathrm{2}^{{n}} }}{\frac{\pi}{\mathrm{2}^{{n}} }}=\pi \\ $$

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