If the value of lim _(x → 0) (2 - cos x√(cos 2x) )^(( (x + 2)/x² )) is equal to eᵃ, then a is equal to __________.

Solution

Maths sir · 4 steps

Base goes to 1 and the power blows up, so this is 1^∞: pull the exponent out with eˡⁱᵐ⁽ᶠ⁻¹⁾ᵍ.

1. Spot the form. As x → 0, the base 2 - cos x√(cos 2x) → 2 - 1 = 1 and the exponent (x+2)/x² → ∞. For a 1^∞ limit of fᵍ, the answer is e raised to the limit of (f-1)g.

   lim(x → 0) fᵍ = e^(lim(x → 0) (f-1) g) = e^(lim(x → 0) (1 - cos x√(cos 2x))(x+2)/x²)

2. Expand near zero. cos x ≈ 1 - x²/2 and cos 2x ≈ 1 - 2x², so √(cos 2x) ≈ 1 - x². Their product is 1 - 3x²/2 up to the x² term, so 1 - cos x√(cos 2x) ≈ 3x²/2.
3. Find the exponent. 3x²/2 × (x+2)/x² = 3(x+2)/2 → (3 × 2)/2 = 3 as x → 0.
4. So. The limit is e³, so a = 3.

The common slip: 5. Expanding √(cos 2x) as 1 - 2x² (forgetting that the square root halves the x² term) gives 1 - f ≈ 5x²/2 and a = 5.

The idea to keep. For any 1^∞ limit, write it as eˡⁱᵐ⁽ᶠ⁻¹⁾ᵍ. Then expand f-1 only up to the power that cancels the blow-up in g.

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