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Question Number 102020 by Dwaipayan Shikari last updated on 06/Jul/20

$$\underset{x\to 0}{\lim }{\left(tan(\frac{\pi }{4}-x)\right)}^{\frac{1}{x}}$$

Answered by john santu last updated on 06/Jul/20

$$\underset{x\to 0}{\lim }{(1+(\tan (\frac{\pi }{4}-x)-1))}^{\frac{1}{x}}$$$$=e^{\underset{x\to 0}{\lim }\left(\frac{\tan (\frac{\pi }{4}-x)-1}{x}\right)}$$$$=e^{\underset{x\to 0}{\lim }\frac{-{\sec }^2(\frac{\pi }{4}-x)}{1}}=e^{-2}=\frac{1}{e^2}$$$$(JS⊛)$$

Answered by Ar Brandon last updated on 06/Jul/20

$$\mathrm{Let}\mathrm{y}=\underset{\mathrm{x}\to 0}{\lim }{\left[\tan (\frac{\pi }{4}-\mathrm{x})\right]}^{\frac{1}{\mathrm{x}}}\Rightarrow \mathrm{lny}=\underset{\mathrm{x}\to 0}{\lim }\left\{\frac{\ln \left[\tan (\frac{\pi }{4}-\mathrm{x})\right]}{\mathrm{x}}\right\}$$$$\Rightarrow \mathrm{lny}=\underset{\mathrm{x}\to 0}{\lim }\{-\sec (\frac{\pi }{4}-\mathrm{x})\mathrm{cosec}(\frac{\pi }{4}-\mathrm{x})\}$$$$=-\sec \left(\frac{\pi }{4}\right)\mathrm{cosec}\left(\frac{\pi }{4}\right)=-2$$$$\Rightarrow \underset{\mathrm{x}\to 0}{\lim }{\left[\tan (\frac{\pi }{4}-\mathrm{x})\right]}^{\frac{1}{\mathrm{x}}}=\frac{1}{{\mathrm{e}}^2}$$

Commented by Dwaipayan Shikari last updated on 06/Jul/20

$$Great$$

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