Let β=lim _(x → 0) (α x-(e³ˣ-1))/(α x(e³ˣ-1)) for some α ∈ ℝ. Then the value of α+β is :

Solution

Maths sir · 3 steps

The denominator vanishes like x², so the numerator's x term must die first. That fixes α.

1. Expand the exponential. Use e³ˣ - 1 = 3x + 9x²/2 + 9x³/2 + … near x = 0. The denominator is then α x · 3x = 3α x² to leading order.
2. Force a finite limit. The numerator is (α - 3)x - 9x²/2 - …. Divided by 3α x², the x term would blow up, so its coefficient must be zero: α = 3.

   β = lim(x → 0) (-9x²/2 - 9x³/2 - …)/3x(3x + 9x²/2 + …) = (-9/2)/9 = -1/2

3. So. α + β = 3 - 1/2 = 5/2, which is option (c).

The idea to keep. When a limit is given as finite but has an unknown constant, expand in powers of x and set every coefficient that would blow up to zero. That condition gives the constant.

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