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Question Number 18627 by Arnab Maiti last updated on 26/Jul/17

$$\underset{\mathrm{x}\to 0}{\lim }{\left(\frac{\sin \mathrm{x}}{\mathrm{x}}\right)}^{\frac{\sin \mathrm{x}}{\mathrm{x}-\sin \mathrm{x}}}$$

Commented by Arnab Maiti last updated on 26/Jul/17

$$\mathrm{Please}\mathrm{help}.$$

Answered by 433 last updated on 26/Jul/17

$$$$$$$$$$\frac{\sin x}{x}=y$$$$x\to 0\Rightarrow y\to 1$$$$\frac{\sin x}{x-\sin x}=\frac{\frac{\sin x}{x}}{\frac{x}{x}-\frac{\sin x}{x}}=\frac{y}{1-y}=-1+\frac{1}{1-y}$$$$\underset{y\to 1}{\lim }\frac{y^{\frac{1}{1-y}}}{y}=\underset{y\to 1}{\lim }\frac{e^{\frac{1}{1-y}\ln \left(y\right)}}{y}=e^{-1}$$$$\frac{\ln y}{1-y}\stackrel{DLH}{=}\frac{\frac{1}{y}}{-1}\stackrel{y\to 1}{\to }-1$$

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