# Find solution lim_(x->0)( tanx-sinx)/x^3

Find solution $\underset{x\to 0}{lim}\frac{\mathrm{tan}x-\mathrm{sin}x}{{x}^{3}}$
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Benedict
$\mathrm{tan}\left(x\right)=x+\frac{{x}^{3}}{3}+2\frac{{x}^{5}}{15}+17\frac{{x}^{7}}{315}+\dots$
$\mathrm{sin}\left(x\right)=x-\frac{{x}^{3}}{3}!+\frac{{x}^{5}}{5}!-\frac{{x}^{7}}{7}!+\dots$
$\mathrm{tan}\left(x\right)-\mathrm{sin}\left(x\right)={x}^{3}\left(\frac{1}{3}+\frac{1}{3}!\right)+{x}^{5}\left(\frac{2}{15}+\frac{1}{5}!\right)+{x}^{7}\left(\frac{17}{315}+\frac{1}{7}!\right)+\dots$
$f\left(x\right)=\frac{\mathrm{tan}\left(x\right)-\mathrm{sin}\left(x\right)}{{x}^{3}}=\frac{1}{2}+17\frac{{x}^{2}}{120}+13\frac{{x}^{4}}{240}+\dots$