If lim x 0 ( ( a - n ) n x - tan x ) sin n x x 2 = 0 , where n is a nonzero real number, then a is equal to [2003]

Question

If $\lim_{x \to 0} \frac{((a - n) n x - \tan x) \sin n x}{x^{2}} = 0,$ where $n$ is a nonzero real number, then $a$ is equal to [2003]

Smart Achievers · Accepted Answer

(4)

$\lim_{x \to 0} \frac{[(a - n) n x - \tan x] \sin n x}{x^{2}} = 0$

$\Rightarrow \lim_{x \to 0} n \cdot \frac{\sin n x}{n x} [{(a - n) n - \frac{\tan x}{x}}] = 0$

$\Rightarrow n \cdot 1 \cdot [(a - n) n - 1] = 0$ $\Rightarrow (a - n) n - 1 = 0$

$\Rightarrow a = \frac{1}{n} + n$ $[∵ n is non zero real number]$

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