# Solve limit <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXAt

Solve limit $\underset{x\to 0}{lim}\frac{x-\mathrm{sin}\left(x\right)}{\left(x\mathrm{sin}\left(x\right){\right)}^{3/2}}$
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Julien Carrillo
You have with Taylor series around 0 :
$\mathrm{sin}\left(x\right)=x-\frac{{x}^{3}}{6}+o\left({x}^{3}\right)$
$\underset{x\to 0}{lim}\frac{x-\mathrm{sin}\left(x\right)}{\left(x\mathrm{sin}\left(x\right){\right)}^{3/2}}=\underset{x\to 0}{lim}\frac{x-x+\frac{{x}^{3}}{6}+o\left({x}^{3}\right)}{\left(x\left(x+o\left(x\right){\right)}^{3/2}}=\underset{x\to 0}{lim}\frac{\frac{{x}^{3}}{6}+o\left({x}^{3}\right)}{{x}^{3}+o\left({x}^{3}\right)}=\frac{1}{6}$
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Mackenzie Rios
For $x\to 0$ we have $\left(\mathrm{sin}x{\right)}^{3/2}\sim {x}^{3/2}$ and so
$\frac{x-\mathrm{sin}x}{\left(x\mathrm{sin}x{\right)}^{3/2}}\sim \frac{\frac{{x}^{3}}{3!}}{{x}^{3/2}\left(x{\right)}^{3/2}}=\frac{1}{6}$