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dokezwa17 2022-05-19 Answered

Solve limit

lim_{x \to 0} \frac{x - \sin (x)}{(x \sin (x))^{3 / 2}}

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Answers (2)

Julien Carrillo

Answered 2022-05-20 Author has 13 answers

You have with Taylor series around 0 :

\sin (x) = x - \frac{x^{3}}{6} + o (x^{3})

lim_{x \to 0} \frac{x - \sin (x)}{(x \sin (x))^{3 / 2}} = lim_{x \to 0} \frac{x - x + \frac{x^{3}}{6} + o (x^{3})}{(x (x + o (x))^{3 / 2}} = lim_{x \to 0} \frac{\frac{x^{3}}{6} + o (x^{3})}{x^{3} + o (x^{3})} = \frac{1}{6}

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Mackenzie Rios

Answered 2022-05-21 Author has 4 answers

For

x \to 0

we have

(\sin x)^{3 / 2} \sim x^{3 / 2}

and so

\frac{x - \sin x}{(x \sin x)^{3 / 2}} \sim \frac{\frac{x^{3}}{3!}}{x^{3 / 2} (x)^{3 / 2}} = \frac{1}{6}

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