The value of limn→∞ cos⁡x2cos⁡x4cos⁡x8…cos⁡x2n, is

# The value of $\underset{n\to \mathrm{\infty }}{lim} \mathrm{cos}\left(\frac{x}{2}\right)\mathrm{cos}\left(\frac{x}{4}\right)\mathrm{cos}\left(\frac{x}{8}\right)\dots \mathrm{cos}\left(\frac{x}{{2}^{n}}\right),$ is

1. A

1

2. B

$\frac{\mathrm{sin}x}{x}$

3. C

$\frac{x}{\mathrm{sin}x}$

4. D

none of these

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### Solution:

We know that

$\mathrm{cos}A\mathrm{cos}2A\mathrm{cos}4A\dots \mathrm{cos}{2}^{n-1}A=\frac{\mathrm{sin}{2}^{n}A}{{2}^{n}\mathrm{sin}A}$

Taking $A=\frac{x}{{2}^{n}},$ we get

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