If $\lim _{x \rightarrow 0} \frac{\operatorname{axe}^{x}-\beta \log _{e}(1+x)+\gamma x^{2} e^{-x}}{x \sin ^{2} x}=10, \alpha, \beta, \gamma \in \mathrm{R}$, then the value of $\alpha+\beta+\gamma$ is.......

JEE Main · 2021 · 20 Jul 2021 · Shift 2 · Limits

lim_{x \to 0} \frac{axe ^{x} - β l o g _{e} ( 1 + x ) + γ x ^{2} e ^{- x}}{x s i n ^{2} x} = 10, α, β, γ \in R

, then the value of

α + β + γ

is.......

Answer: 3

Expanding the numerator in powers of

x

α x e^{x} = α (x + x^{2} + \frac{x ^{3}}{2} + \dots), β lo g_{e} (1 + x) = β (x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots), γ x^{2} e^{- x} = γ (x^{2} - x^{3} + \dots)

while the denominator is

x sin^{2} x = x^{3} + O (x^{5})

. A finite limit forces the coefficients of

x

and

x^{2}

in the numerator to vanish, and the coefficient of

x^{3}

to equal

10

α - β = 0, α + \frac{β}{2} + γ = 0, \frac{α}{2} - \frac{β}{3} - γ = 10

The first two give

β = α

and

γ = - \frac{3 α}{2}

. Substituting in the third,

\frac{α}{2} - \frac{α}{3} + \frac{3 α}{2} = \frac{5 α}{3} = 10 ⟹ α = 6, β = 6, γ = - 9

α + β + γ = 3

JEE Main · 2021 · 20 Jul 2021 · Shift 2 · Integer answer