The value of lim n → ∞ cos ( x/2 ) cos (x/4 ) cos ( x/8 )......cos( x/2n  ) is : (a) x╱sinx ¸ (b) x╱cosx ¸ (c) sinx╱x ¸ (d) cosx╱x

Asked by yeeshuraj3 | 15th Sep, 2016, 05:24: PM

Expert Answer:

sin x space equals space 2 to the power of 1 sin x over 2 cos x over 2
sin x space equals space 2 squared sin x over 4 cos x over 4 cos x over 2
sin x space equals space 2 cubed sin x over 8 cos x over 8 cos x over 4 cos x over 2
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sin x space equals space 2 to the power of n sin x over 2 to the power of n begin inline style product from k equals 1 to n of end style cos x over 2 to the power of k

We space know space that
sinx over straight x rightwards arrow 1 space space space as space space straight x rightwards arrow 0


limit as straight x rightwards arrow infinity of begin inline style product from straight k equals 1 to straight n of end style begin inline style cos end style begin inline style straight x over 2 to the power of straight k end style space equals space limit as straight x rightwards arrow infinity of sinx over straight x open parentheses fraction numerator sin 2 to the power of negative straight n end exponent straight x over denominator 2 to the power of negative straight n end exponent straight x end fraction close parentheses to the power of negative 1 end exponent equals fraction numerator sin space straight x over denominator straight x end fraction

Answered by Vijaykumar Wani | 16th Sep, 2016, 06:44: PM