If lim _(x → 0) (cos (2x) + a cos (4x) - b)/x⁴is finite, then (a + b) is equal to:

Solution

Maths sir · 4 steps

Expand both cosines up to x⁴, then kill the x⁰ and x² terms so nothing blows up.

1. Expand the cosines. Use cos θ = 1 - θ²/2 + θ⁴/24 - … with θ = 2x and θ = 4x. This gives cos 2x = 1 - 2x² + 2/3x⁴ - … and cos 4x = 1 - 8x² + 32/3x⁴ - …

   cos 2x + a cos 4x - b = (1 + a - b) - (2 + 8a)x² + (2/3 + 32a/3)x⁴ + …

2. Condition for finiteness. After dividing by x⁴, the constant term and the x² term would go to infinity as x → 0. So both coefficients must be zero: 2 + 8a = 0 gives a = -1/4, and 1 + a - b = 0 gives b = 3/4.
3. Check the limit. The limit is then the x⁴ coefficient: 2/3 + 32/3(-1/4) = 2/3 - 8/3 = -2, which is finite.
4. So. a + b = -1/4 + 3/4 = 1/2, which is option (d).

The idea to keep. For a limit over xⁿ to be finite, every term of the numerator below xⁿ must vanish. Expand with Taylor series and set those coefficients to zero.

Answer from the NTA final answer key. Solution by PrepWiser.