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Question Number 131033 by greg_ed last updated on 31/Jan/21

$$\underset{x\to 0}{\lim }\frac{x-\tan x}{x{\tan }^{2}x}=?$$

Commented by greg_ed last updated on 02/Feb/21

$$\mathbf{without}\mathbf{the}\mathbf{limited}\mathbf{development}\mathbf{method}!$$

Answered by Dwaipayan Shikari last updated on 31/Jan/21

$$\underset{x\to 0}{\lim }\frac{xcosx-sinx}{{xsin}^{2}x}=\frac{x(1-\frac{x^2}{2})-x+\frac{x^3}{6}}{x^3}=-\frac{x}{3x}=-\frac{1}{3}$$

Answered by mathmax by abdo last updated on 31/Jan/21

$$\mathrm{we}\mathrm{have}\mathrm{tanx}\sim \mathrm{x}+\frac{{\mathrm{x}}^3}{3}\Rightarrow \mathrm{x}-\mathrm{tanx}\sim -\frac{{\mathrm{x}}^3}{3}\mathrm{also}{\mathrm{xtan}}^2\mathrm{x}\sim {\mathrm{x}}^3\Rightarrow$$$$\frac{\mathrm{x}-\mathrm{tanx}}{{\mathrm{xtan}}^2\mathrm{x}}\sim -\frac{{\mathrm{x}}^3}{{3\mathrm{x}}^3}=-\frac{1}{3}\Rightarrow {\lim }_{\mathrm{x}\to 0}\frac{\mathrm{x}-\mathrm{tanx}}{{\mathrm{xtan}}^2\mathrm{x}}=-\frac{1}{3}$$

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