If lim _(x → 0) (α xeˣ - β log ₑ(1 + x) + γ x²e⁻ˣ)/(xsin ²x) = 10,α ,β ,γ ∈ R, then the value of α + β + γ is _____________.

Solution

Maths sir · 4 steps

The denominator behaves like x³, so expand the numerator up to x³ and match coefficients.

1. Size of denominator. Near 0, sin x ≈ x, so x sin² x ≈ x³. For a finite limit, the x and x² terms of the numerator must vanish. The x³ coefficient must then equal 10.

   α x eˣ - β ln(1+x) + γ x² e⁻ˣ = (α-β)x + (α+β/2+γ)x² + (α/2-β/3-γ)x³ + …

2. Kill low powers. From x: α - β = 0, so β = α. From x²: α + α/2 + γ = 0, so γ = -3α/2.
3. Match the limit. Now set the x³ coefficient to 10: α/2 - α/3 + 3α/2 = 5α/3 = 10. This gives α = 6, so β = 6 and γ = -9.
4. So. α + β + γ = 6 + 6 - 9 = 3.

The idea to keep. When a limit is finite and nonzero, find the lowest power in the denominator. Expand the numerator to that power, set every lower coefficient to zero, and match the last one to the limit.

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