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Question Number 122106 by bemath last updated on 14/Nov/20

\lim_{x \to 0} \frac{3 \sin (π x) - \sin (3 π x)}{x^{3}} = ?

Answered by liberty last updated on 14/Nov/20

\lim_{x \to 0} \frac{3 (π x - \frac{π^{3} x^{3}}{6}) - (3 π x - \frac{27 π^{3} x^{3}}{6})}{x^{3}}

\lim_{x \to 0} \frac{- \frac{π^{3} x^{3}}{2} + \frac{9 π^{3} x^{3}}{2}}{x^{3}} = 4 π^{3} . ▴

Answered by Bird last updated on 14/Nov/20

f (x) = \frac{3 s i n (π x) - s i n (3 π x)}{x^{3}}

\Rightarrow f (x) =_{π x = t} π^{3} \times \frac{3 s i n (t) - s i n (3 t)}{t^{3}}

w e h a v e s i n t \sim t - \frac{t^{3}}{6}

s i n (3 t) \sim 3 t - \frac{27 t^{3}}{6} = 3 t - \frac{9}{2} t^{3}

\Rightarrow f (\frac{t}{π}) \sim π^{3} . \frac{3 t - \frac{t^{3}}{2} - 3 t + \frac{9}{2} t^{3}}{t^{3}}

\Rightarrow f (\frac{t}{π}) \sim 4 π^{3} \Rightarrow {l i m}_{x \to 0} f (x) = 4 π^{3}

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