Let be such that lim x 0 x 2 sin ( x ) x - sin x = 1 . Then 6 equals ______. [2016]

Question

Let $α, β \in ℝ$ be such that $\lim_{x \to 0} \frac{x^{2} \sin (β x)}{α x - \sin x} = 1 .$ Then $6 (α + β)$ equals ______. [2016]

Smart Achievers · Accepted Answer

(7)

$\lim_{x \to 0} \frac{x^{2} \sin (β x)}{α x - \sin x} = 1$

$\Rightarrow \lim_{x \to 0} \frac{x^{3} β}{α x - \sin x} = 1$

$\Rightarrow \lim_{x \to 0} \frac{x^{3} β}{α x - (x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots \infty)} = 1$

$\Rightarrow \lim_{x \to 0} \frac{x^{3} β}{(α - 1) x + \frac{x^{3}}{3!} - \frac{x^{5}}{5!} + \dots \infty} = 1$

It is possible when

$α - 1 = 0$ and $β = \frac{1}{3!}$

$\Rightarrow α = 1$ and $β = \frac{1}{6}$

$∴$ $6 (α + β) = 6 (1 + \frac{1}{6}) = 7$

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