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Question Number 213821 by hardmath last updated on 17/Nov/24

$$\mathrm{Find}:\underset{\mathbf{x}\to 0}{\lim }{\left(\frac{\mathrm{sinx}}{\mathrm{x}}\right)}^{\frac{\mathrm{sinx}}{\mathrm{x}-\mathrm{sinx}}}=?$$

Answered by mehdee7396 last updated on 17/Nov/24

$${lim}_{x\to 0}(\frac{sinx}{x}-1)\frac{sinx}{x-sinx}$$$$={lim}_{x\to 0}\left(\frac{sinx-x}{x}\right)\frac{sinx}{x-sinx}=-1$$$$\Rightarrow answer=e^{-1}$$$$$$

Answered by lepuissantcedricjunior last updated on 20/Nov/24

$$\underset{x\to 0}{\lim }{\left(\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{\mathit{x}}\right)}^{\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{\mathit{x}-\mathit{s}\mathit{i}\mathit{n}\mathit{x}}}=1^{\mathit{F}\mathit{I}}$$$$=>\underset{x\to 0}{\lim }{\left(\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{\mathit{x}}\right)}^{\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{\mathit{x}-\mathit{s}\mathit{i}\mathit{n}\mathit{x}}}=\underset{x\to 0}{\lim }{\mathit{e}}^{\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{l}\mathit{n}\left(\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{\mathit{x}}\right)}{\mathit{x}-\mathit{s}\mathit{i}\mathit{n}\mathit{x}}}$$$$\mathit{d}\mathit{l}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\stackrel{`}{\mathit{a}}{\mathit{l}}^{\prime }o\mathit{r}\mathit{d}\mathit{r}\mathit{e}3\mathit{a}\mathit{u}\mathit{v}\mathit{o}\mathit{i}\mathit{s}\mathit{d}\mathit{e}0$$$$\mathit{s}\mathit{i}\mathit{n}\mathit{x}=\underset{\mathit{k}=0}{\sum ^{\mathit{n}}}\frac{{\mathit{x}}^{\mathit{k}}}{\mathit{k}!}\times {\mathit{f}}^{\mathit{k}}\left(0\right)=\mathit{x}-\frac{{\mathit{x}}^3}{6}+\mathit{x}{\left(0\right)}^3$$$$\underset{x\to 0}{\lim }{\mathit{e}}^{\frac{(\mathit{x}-\frac{{\mathit{x}}^3}{6})\mathit{l}\mathit{n}(1-\frac{{\mathit{x}}^2}{6})}{\mathit{x}-\mathit{x}+\frac{{\mathit{x}}^3}{6}}}\mathit{o}\mathit{r}\mathit{d}\mathit{l}\mathit{l}\mathit{n}(1-\mathit{t})\stackrel{`}{\mathit{a}}{\mathit{l}}^{\prime }\mathit{o}\mathit{r}\mathit{d}\mathit{r}\mathit{e}3\mathit{a}\mathit{u}\mathit{v}\mathit{o}\mathit{i}\mathit{s}\mathit{d}\mathit{e}0\mathit{d}\mathit{o}\mathit{n}\mathit{n}\mathit{e}$$$$\mathit{l}\mathit{n}(1-\mathit{t})=-\mathit{t}+\frac{{\mathit{t}}^2}{2}-\frac{{\mathit{t}}^3}{3}+\mathit{t}{\left(0\right)}^3$$$$=>\underset{x\to 0}{\lim }{\left(\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{\mathit{x}}\right)}^{\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{\mathit{x}-\mathit{s}\mathit{i}\mathit{n}\mathit{x}}}=\underset{x\to 0}{\lim }{\mathit{e}}^{\frac{(\mathit{x}-\frac{{\mathit{x}}^3}{6})(-\frac{{\mathit{x}}^2}{6}+\frac{{\mathit{x}}^4}{12})}{\frac{{\mathit{x}}^3}{6}}}$$$$=\underset{x\to 0}{\lim }{\mathit{e}}^{\frac{-\frac{{\mathit{x}}^3}{6}}{\frac{{\mathit{x}}^3}{6}}}={\mathit{e}}^{-1}$$$$..........................................................$$$$profcedricjunior.....$$

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