If limx→0((a−n)nx−tanx)sinnxx2=0 where n is nonzero

Question -

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

Options:

A .

0

B .

\frac{n + 1}{n}

C .

n

D .

n + \frac{1}{n}

Answer: Option D
:
D

lim x \to 0 \frac{((a - n) n x - tan x) sin n x}{x^{2}} = 0

lim x \to 0 ((a - n) n - (\frac{tan x}{x})) . \frac{sin n x}{n x} . n = 0

\Rightarrow ((a - n) n - 1) .1 . n = 0

\Rightarrow a = \frac{1}{n} + n

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