Solve limit $\underset{x\to 0}{lim}\frac{x-\mathrm{sin}(x)}{(x\mathrm{sin}(x){)}^{3/2}}$

dokezwa17
2022-05-19
Answered

Solve limit $\underset{x\to 0}{lim}\frac{x-\mathrm{sin}(x)}{(x\mathrm{sin}(x){)}^{3/2}}$

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Julien Carrillo

Answered 2022-05-20
Author has **13** answers

You have with Taylor series around 0 :

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

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

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

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

Mackenzie Rios

Answered 2022-05-21
Author has **4** answers

For $x\to 0$ we have $(\mathrm{sin}x{)}^{3/2}\sim {x}^{3/2}$ and so

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

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

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