lim _(x → 0) (cos (sin x) - cos x)/x⁴ is equal to :

Solution

Maths sir · 4 steps

Turn the difference of cosines into a product of sines, then each factor has a standard limit.

1. Use C minus D. The form is 0/0. Apply 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)/x·sin((x-sin x)/2)/x³

2. First factor. Near 0, sin(x+sin x)/2 ≈ (x+sin x)/2. Dividing by x gives 1/2(1 + (sin x)/x) → 1/2(1+1) = 1.
3. Second factor. Since sin x = x - x³/6 + …, we get x - sin x ≈ x³/6. So sin(x-sin x)/2 ≈ x³/12, and dividing by x³ gives 1/12.
4. So. The limit is 2 × 1 × 1/12 = 1/6, which is option (c).

The idea to keep. A difference of cosines over a power of x becomes easy once you turn it into a product of sines. Remember x - sin x ≈ x³/6 near 0, because it comes up again and again.

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