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Question Number 165054 by mathlove last updated on 25/Jan/22

Answered by alephzero last updated on 25/Jan/22

$$\underset{x\to 2}{\lim }{(\sqrt{\sin \frac{\pi }{x}}:\cos \frac{\pi }{x+2})}^{\frac{1}{x}}=$$$$=\sqrt{\sqrt{\sin \frac{\pi }{2}}:\cos \frac{\pi }{4}}=$$$$=\sqrt{1:\frac{\sqrt{2}}{2}}=\sqrt{\frac{2}{\sqrt{2}}}=\frac{\sqrt{2}}{\sqrt{\sqrt{2}}}=\frac{\sqrt{2}}{\sqrt[4]{2}}=$$$$=\frac{\sqrt{2}\sqrt[4]{2^3}}{2}=\frac{\sqrt[4]{2^5}}{2}=\frac{2\sqrt[4]{2}}{2}=\sqrt[4]{2}$$

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