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Question Number 25588 by behi.8.3.4.17@gmail.com last updated on 11/Dec/17

$$\underset{x\to 0}{\lim }\left(\frac{\mathit{x}-\mathit{t}\mathit{g}\mathit{x}}{\mathit{x}-\mathit{s}\mathit{i}\mathit{n}\mathit{x}}\right)=?$$$${\mathit{l}}^{\prime }\mathit{h}\mathit{o}\mathit{p}\mathit{i}\mathit{t}\mathit{a}\mathit{l}\mathit{r}\mathit{o}\mathit{l}\mathit{e}\mathit{n}\mathit{o}\mathit{t}\mathit{a}\mathit{l}\mathit{l}\mathit{o}\mathit{w}\mathit{e}\mathit{d}!$$

Commented by moxhix last updated on 12/Dec/17

https://math.stackexchange.com/questions/508733/lim-x-to0-fracx-sin-xx-tan-x-without-using-lhopital .

Answered by kaivan.ahmadi last updated on 13/Dec/17

$$\mathrm{we}\mathrm{know}\mathrm{that}(\mathrm{when}\mathrm{x}\to 0)$$$$\mathrm{sinx}\approx \mathrm{x}-{\mathrm{x}}^3/6$$$$\mathrm{and}$$$$\mathrm{tanx}\approx \mathrm{x}+{\mathrm{x}}^3/3$$$$\mathrm{so}$$$$$$$$\frac{\mathrm{x}-\mathrm{tanx}}{\mathrm{x}-\mathrm{sinx}}\approx \frac{{\mathrm{x}}^3/6}{-{\mathrm{x}}^3/3}=-\frac{1}{2}$$

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