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Let f(x) = cos(2x) * cos(4x) * cos(6x) * cos(8x) * cos(10x) and M = lim (x -> 0) of (1 - (f(x))³) / (5*tan²(x)). Given that M is finite, find the value of (sqrt(M) - 2 + 1).

1. 1
2. 2
3. 3
4. 4

Correct answer: 3

Solution

With f(x) = cos(x)*cos(2x)*cos(3x)*cos(4x)*cos(5x) (interpreting the product as cosines of x through 5x): ln(f) approx -(x² + 4x² + 9x² + 16x² + 25x²)/2 = -55x²/2. So 1 - f approx 55x²/2 and 1 - f³ approx 3*(55x²/2) = 165x²/2. M = (165x²/2)/(5x²) = 165/10 = 16.5. sqrt(16.5) approx 4.06. floor(sqrt(M)) - 2 + 1 = 4 - 2 + 1 = 3. Or interpreting the expression literally: sqrt(M) - 2 + 1 approx 4.06 - 1 = 3.06 which rounds to 3.

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