If the function f(x) = (cos (sin x) - cos x)/x⁴ is continuous at each point in its domain and f(0) = 1/k, then k is ____________.

Solution

Maths sir · 5 steps

Turn the difference of cosines into a product of sines, and each sine factor has a limit you already know.

1. Continuity condition. For f to be continuous at 0, f(0) must equal lim(x → 0) (cos(sin x) - cos x)/x⁴. So find this limit.
2. Sum to product. Use cos A - cos B = 2 sin(A+B)/2 sin(B-A)/2 with A = sin x and B = x.

   (cos(sin x) - cos x)/x⁴ = (2 sin((x+sin x)/2)sin((x-sin x)/2))/x⁴

3. Small x behaviour. As x → 0, sin θ ≈ θ for each small angle. Also x + sin x ≈ 2x, and x - sin x ≈ x³/6 because sin x = x - x³/6 + ….
4. Put together. The numerator behaves like 2 × x × x³/12 = x⁴/6. Dividing by x⁴ gives the limit 1/6.
5. So. f(0) = 1/k = 1/6, so k = 6.

The common slip: 12. Dropping the factor 2 in front of the sum-to-product formula gives the limit 1/12, so k = 12 instead of 6.

The idea to keep. For a difference of two cosines over a power of x, convert it to a product of sines. Then use sin θ ≈ θ and the series x - sin x ≈ x³/6 for each factor.

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