The value of lim x - 0lim n -cosx-2cosx-22cosx-23-cosx-2n is

L=lim_x→0lim_n→∞(cosx2cosx2^2cosx2^3⋯cosx2^n) =lim_x→0lim_n→∞(cosx2^ncosx2^n 1cosx2^n 2⋯cosx2^2cosx2) =lim_x→0lim_n→∞[12sinx2^n(2sinx2^ncosx2^n)cosx2^n 1cosx2^n ...

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Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

The value of ...

Question

The value of $lim x \to 0 lim n \to \infty (cos \frac{x}{2} cos \frac{x}{2^{2}} cos \frac{x}{2^{3}} \dots cos \frac{x}{2^{n}})$ is

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Solution

$L = lim x \to 0 lim n \to \infty (cos \frac{x}{2} cos \frac{x}{2^{2}} cos \frac{x}{2^{3}} \dots cos \frac{x}{2^{n}})$
$= lim x \to 0 lim n \to \infty (cos \frac{x}{2^{n}} cos \frac{x}{2^{n - 1}} cos \frac{x}{2^{n - 2}} \dots cos \frac{x}{2^{2}} cos \frac{x}{2})$
$= lim x \to 0 lim n \to \infty ⎡ ⎢ ⎢ ⎣ \frac{1}{2 sin \frac{x}{2^{n}}} (2 sin \frac{x}{2^{n}} cos \frac{x}{2^{n}}) cos \frac{x}{2^{n - 1}} cos \frac{x}{2^{n - 2}} \dots cos \frac{x}{2^{2}} cos \frac{x}{2} ⎤ ⎥ ⎥ ⎦$
$= lim x \to 0 lim n \to \infty ⎡ ⎢ ⎢ ⎣ \frac{1}{2^{2} sin \frac{x}{2^{n}}} (2 sin \frac{x}{2^{n - 1}} cos \frac{x}{2^{n - 1}}) cos \frac{x}{2^{n - 2}} \dots cos \frac{x}{2^{2}} cos \frac{x}{2} ⎤ ⎥ ⎥ ⎦$
$= lim x \to 0 lim n \to \infty ⎡ ⎢ ⎢ ⎣ \frac{1}{2^{3} sin \frac{x}{2^{n}}} (2 sin \frac{x}{2^{n - 2}} cos \frac{x}{2^{n - 2}}) \dots cos \frac{x}{2^{2}} cos \frac{x}{2} ⎤ ⎥ ⎥ ⎦$
$= lim x \to 0 lim n \to \infty ⎡ ⎢ ⎢ ⎣ \frac{1}{2^{n} sin \frac{x}{2^{n}}} \times (2 sin \frac{x}{2} cos \frac{x}{2}) ⎤ ⎥ ⎥ ⎦$
$= lim x \to 0 lim n \to \infty \frac{sin x}{2^{n} sin \frac{x}{2^{n}}}$
$= lim x \to 0 ⎡ ⎢ ⎢ ⎣ lim n \to \infty \frac{sin x}{x} \times \frac{\frac{x}{2^{n}}}{sin \frac{x}{2^{n}}} ⎤ ⎥ ⎥ ⎦$
$= lim x \to 0 (\frac{sin x}{x}) \times 1 (as n \to \infty, \frac{x}{2^{n}} \to 0)$
$= 1$

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The value of limx→0limn→∞(cosx2cosx22cosx23⋯cosx2n) is

The value of $lim x \to 0 lim n \to \infty (cos \frac{x}{2} cos \frac{x}{2^{2}} cos \frac{x}{2^{3}} \dots cos \frac{x}{2^{n}})$ is