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Limit of 1 minus a product of cosines (or log of a product of secants) over x^2

This type of question, asked 3 times: 8 Apr 2024, 28 Jan 2026 S1, 5 Apr 2026 S1. All 3 questions below, oldest first.

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  1. Question 1

    8 Apr 2024

    90
    The value of lim⁡x→02(1−cos⁡xcos⁡2xcos⁡3x3…..cos⁡10x10x2)\lim_{x\rightarrow 0} 2\left( \frac{1 - \cos x\sqrt{\cos2x}\sqrt[3]{\cos3x}\ldots..\sqrt[10]{\cos10x}}{x^{2}} \right) is

  2. Question 2

    28 Jan 2026 S1

    64
    The value of lim⁡x→0log⁡e(sec(ex)⋅sec(e2x)⋅…⋅sec(e10x))e2−e2cos⁡x\underset{x\rightarrow 0}{\lim} \frac{\log_{e}\left( sec(ex) \cdot sec\left( e^{2}x \right)\cdot \ldots \cdot sec\left( e^{10}x \right) \right)}{e^{2}-e^{2\cos x}} is equal to

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  3. Question 3

    5 Apr 2026 S1

    67
    The product of all possible values of α\alpha, for which lim⁡x→0(1−cos⁡(αx)cos⁡((α+1)x)cos⁡((α+2)x)sin⁡2((α+1)x))=2\lim_{x\rightarrow 0} \left( \frac{1 - \cos(\alpha x)\cos((\alpha+ 1)x)\cos((\alpha+ 2)x)}{\sin^{2}((\alpha+ 1)x)} \right)= 2, is :

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