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Question Number 188449 by mnjuly1970 last updated on 01/Mar/23

$$$$$$\mathrm{calculate}$$$$$$$${\lim }_{x\to \frac{\pi }{4}}{\left(\tan \left(x\right)\right)}^{\tan \left(2x\right)}=?$$$$$$

Answered by cortano12 last updated on 01/Mar/23

$$\mathrm{L}=\underset{x\to \frac{\pi }{4}}{\lim }{\left[\tan \mathrm{x}\right]}^{\tan 2\mathrm{x}}$$$$\mathrm{L}={\mathrm{e}}^{\underset{x\to \frac{\pi }{4}}{\lim }\left(\frac{\tan \mathrm{x}-1}{\tan 2\mathrm{x}}\right)}$$$$\mathrm{L}={\mathrm{e}}^{\underset{x\to \frac{\pi }{4}}{\lim }\frac{\sin \mathrm{x}-\cos \mathrm{x}}{\cos \mathrm{x}}.\frac{\sin 2\mathrm{x}}{(\cos \mathrm{x}-\sin \mathrm{x})(\cos \mathrm{x}+\sin \mathrm{x})}}$$$$\mathrm{L}={\mathrm{e}}^{\underset{x\to \frac{\pi }{4}}{\lim }\left[\frac{-\sin 2\mathrm{x}}{\cos \mathrm{x}(\cos \mathrm{x}+\sin \mathrm{x})}\right]}$$$$\mathrm{L}={\mathrm{e}}^{-\left(\frac{1}{\frac{1}{\sqrt{2}}\left(\sqrt{2}\right)}\right)}=\frac{1}{\mathrm{e}}$$

Commented by mnjuly1970 last updated on 03/Mar/23

$$thxalotsircortano$$

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