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Question Number 80000 by malwaan last updated on 30/Jan/20

$$\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{t}\mathit{h}\mathit{a}\mathit{t}$$$$\underset{\mathit{x}\to 0}{\mathit{l}\mathit{i}\mathit{m}}\frac{\mathit{a}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{x}}{\sqrt{1-{\mathit{x}}^2}}}{\mathit{l}\mathit{n}(1-\mathit{x})}=-1$$

Commented by jagoll last updated on 30/Jan/20

$$\sin y=\frac{x}{\sqrt{1-x^2}}\Rightarrow \frac{x^2}{1-x^2}=\sin {}^{2}y$$$$\frac{1-x^2-1}{1-x^2}=\sin {}^{2}y\Rightarrow 1-\frac{1}{1-x^2}=\sin {}^{2}y$$$$\cos {}^{2}y=\frac{1}{1-x^2}\Rightarrow 1-x^2=\sec {}^{2}y$$$$1-x=\frac{\sec {}^{2}y}{1+x}$$$$$$

Commented by abdomathmax last updated on 30/Jan/20

$$letf\left(x\right)=\frac{arcsin\left(\frac{x}{\sqrt{1-x^2}}\right)}{ln(1-x)}changementx=sint$$$$givef\left(x\right)=\frac{arcsin\left(tant\right)}{ln(1-sint)}=g\left(t\right)$$$$x\to 0\Rightarrow t\to 0\Rightarrow g\left(t\right)\sim \frac{arcsint}{-t}=-\frac{arcsint}{t}$$$$and{lim}_{t\to 0}\frac{arsin\left(t\right)}{t}={arcsin}^{{}^{\prime }}\left(0\right)wehave$$$${arcsin}^{{}^{\prime }}\left(t\right)=\frac{1}{\sqrt{1-t^2}}\Rightarrow {arcsin}^{{}^{\prime }}\left(0\right)=1\Rightarrow$$$${lim}_{t\to 0}g\left(t\right)=-1={lim}_{x\to 0}f\left(x\right)$$

Answered by Kamel Kamel last updated on 30/Jan/20

$$\mathrm{\Omega }=\underset{\mathrm{x}\to 0}{\lim }\frac{\arcsin \left(\frac{\mathrm{x}}{\sqrt{1-{\mathrm{x}}^2}}\right)}{\mathrm{Ln}(1-\mathrm{x})}$$$$\mathrm{Put}:\mathrm{x}=\sin \left(\mathrm{t}\right),\mathrm{x}\to 0\Rightarrow \mathrm{t}\to 0$$$$\mathrm{\Omega }=\underset{\mathrm{x}\to 0}{\lim }\frac{\arcsin \left(\frac{\mathrm{x}}{\sqrt{1-{\mathrm{x}}^2}}\right)}{\mathrm{Ln}(1-\mathrm{x})}=\underset{\mathrm{t}\to 0}{\lim }\frac{\arcsin \left(\tan \left(\mathrm{t}\right)\right)}{\mathrm{Ln}(1-\sin \left(\mathrm{t}\right))}=-\lim \frac{\tan \left(\mathrm{t}\right)}{\sin \left(\mathrm{t}\right)}=-1$$

Commented by malwaan last updated on 31/Jan/20

$$thankyousomuch$$

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