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\lim_{\infty}(\tan x)^{\tan 2 x}

Option: 1

e^{-1}


Option: 2

e


Option: 3

e^3


Option: 4

e^4


Answers (1)

best_answer

\mathrm{ \begin{aligned} & \text { Put } \begin{aligned} & y=(\tan x)^{\tan 2 x} \\ & \Rightarrow \quad \ln y=(\tan 2 x) \ln (\tan x) \\ & \therefore \quad \lim _{x \rightarrow \pi / 4} \ln y=\lim _{x \rightarrow \pi / 4}\left\{\frac{\ln (\tan x)}{\cot 2 x}\right\} \\ &=\lim _{x \rightarrow \pi / 4}\left\{\frac{\sec ^2 x}{-\tan x \times\left(\operatorname{cosec}^2 2 x\right) \times 2}\right\} \\ &=-\frac{2}{1 \times 1 \times 2}=-1 \\ & \because \ln y \rightarrow-1 \text { as } x \rightarrow \pi / 4 \\ & \therefore y \rightarrow e^{-1} \text { as } x \rightarrow \pi / 4 . \end{aligned} \end{aligned} }

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Ajit Kumar Dubey

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