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Question Number 188819 by mnjuly1970 last updated on 07/Mar/23

$$$$$$\mathrm{calculate}$$$$$$$${\lim }_{x\to 0^{+}}{\left(\sqrt{\cos \left(\sqrt{x}\right)}\right)}^{\cot \left(x\right)}=?$$$$$$$$$$

Commented by mehdee42 last updated on 07/Mar/23

$$$$$$\mathrm{calculate}$$$$$$$${\lim }_{x\to 0^{+}}{\left(\sqrt{\cos \left(\sqrt{x}\right)}\right)}^{\cot \left(x\right)}=?$$$$$$$$$$$$notic$$$$if{lim}_{x\to a}f\left(x\right)=1\mathrm{\&}{lim}_{x\to a}g\left(x\right)=\mathrm{\infty }$$$$\Rightarrow {lim}_{x\to a}f{\left(x\right)}^{g\left(x\right)}=e^{{lim}_{x\to a}(f\left(x\right)-1)g\left(x\right)}$$$${lim}_{x\to 0^{+}}\frac{\sqrt{cos\sqrt{x}}-1}{tanx}\stackrel{hop}{=}{lim}_{x\to a}\frac{\frac{-sin\sqrt{x}}{2\sqrt{x}}}{2(1+tanx)\sqrt{cos\sqrt{x}}}=-\frac{1}{4}$$$$\Rightarrow answris:\frac{1}{\sqrt[4]{e}}$$$$$$

Answered by mahdipoor last updated on 07/Mar/23

$$y={\left(\sqrt{cos\left(\sqrt{x}\right)}\right)}^{cot\left(x\right)}={\left[cos\left(\sqrt{x}\right)\right]}^{\frac{cot\left(x\right)}{2}}$$$$\Rightarrow ln\left(y\right)=\frac{cot\left(x\right)}{2}ln\left(cos\left(\sqrt{x}\right)\right)=\frac{ln\left(cos\left(\sqrt{x}\right)\right)}{2tan\left(x\right)}$$$$\Rightarrow x\to 0^{+}lim\left[ln\left(y\right)\right]=lim\frac{ln\left(cos\sqrt{x}\right)}{2tan\left(x\right)}$$$$\Rightarrow =\frac{0}{0},Hop\Rightarrow =\frac{\frac{-sin\left(\sqrt{x}\right)}{cos\left(\sqrt{x}\right)}.\frac{1}{2\sqrt{x}}}{\frac{2}{{cos}^2\left(x\right)}}=$$$$\frac{-{cos}^2\left(x\right)}{4cos\left(\sqrt{x}\right)}\times \left[\frac{sin\left(\sqrt{x}\right)}{\sqrt{x}}\right]=\frac{-1}{4}\times \frac{sin\sqrt{x}}{\sqrt{x}}$$$$\Rightarrow hop\Rightarrow \frac{cos\left(\sqrt{x}\right)\times \frac{1}{2\sqrt{x}}}{\frac{1}{2\sqrt{x}}}=cos\sqrt{x}=1$$$$\Rightarrow \Rightarrow limln\left(y\right)=\frac{-1}{4}\times 1\Rightarrow limy=\frac{1}{{}^4\sqrt{e}}$$$$...............Note:$$$$getlimx\to c\frac{f\left(x\right)}{g\left(x\right)}=,f\left(c\right)=g\left(c\right)=0$$$$\frac{f\left(x\right)}{g\left(x\right)}=k\left(x\right)\frac{u\left(x\right)}{v\left(x\right)}thatk\left(c\right)\ne 0,u\left(c\right)=v\left(c\right)=0$$$$\Rightarrow hop\Rightarrow \frac{k^{{}^{\prime }}\left(c\right)u\left(c\right)+k\left(c\right)u^{{}^{\prime }}\left(c\right)}{v^{{}^{\prime }}\left(c\right)}=k\left(c\right)\frac{u^{{}^{\prime }}\left(c\right)}{v^{{}^{\prime }}\left(c\right)}$$$$...............$$

Commented by mnjuly1970 last updated on 07/Mar/23

$$sepasostad$$

Answered by qaz last updated on 07/Mar/23

$$\underset{x\to 0^{+}}{lim}{\left(\sqrt{\cos \sqrt{x}}\right)}^{\cot x}=\underset{x\to 0^{+}}{lim}{(1-\frac{1}{4}x+o\left(x\right))}^{\frac{1}{x+o\left(x\right)}}=e^{-4}$$

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